Quantum chemical computation of infrared spectra of acidic zeolites Meijer, E.L. DOI: 10.6100/IR537045 Published: 01/01/2000 Document Version Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 08. Aug. 2018
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Quantum chemical computation of infrared spectra ofacidic zeolitesMeijer, E.L.
DOI:10.6100/IR537045
Published: 01/01/2000
Document VersionPublisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differencesbetween the submitted version and the official published version of record. People interested in the research are advised to contact theauthor for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ?
Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.
[4] A. G. Pelmenschikov, R. A. van Santen, J. J�anchen, and E. L. Meijer;
J. Phys. Chem., \CD3CN as a Probe of Lewis and Br�nsted Acidity of
Zeolites", 97, 11071{11074 (1993). The infrared spectra shown in this
16 Chapter 1: Introduction
chapter were taken from this paper. All spectra have been measured by
Jos van Wolput, who also kindly provided the data for the �gures 4 and
5 in this chapter.
[5] C. Paz�e, S. Bordiga, C. Lamberti, M. Salvalaggio, A. Zecchina, and
G. Bellussi; J. Phys. Chem. B, \Acididc Properties of H-� Zeolite As
Probed by Bases with Proton AÆnity in the 118{204 kcal mol�1 Range:
A FTIR Investigation", 101, 4740{4751 (1997).
[6] J. C. Evans, and N. Wright; Spectrochim. Act., \A Peculiar E�ect in the
Infrared Spectra of Certain Molecules", 16, 352{257 (1960).
2Theory of the Computation of Infrared Spectra
Introduction
I have written a computer program `AnharmND' to compute infrared spec-
tra taking into account anharmonicities. This chapter provides some of the
theoretical background on which the computations of the program are based.
In AnharmND solutions for the time-independent Schr�odinger equation
for a limited set of coordinates are approximated. The coordinates are linear
combinations of cartesian atomic coordinates, the kinetic energy is expressed
in terms of products of the conjugated momenta of the coordinates, and the
potential energy is represented by a polynomial in the coordinates. The dipole
surface that is needed for the computation of infrared absorption intensities is
represented by polynomials for each component. The Schr�odinger equation is
solved with a variational approach where the wave functions are expanded in
a basis set of products of Hermite functions. The next section of this chapter
discusses how matrix elements of the Hamiltonian are computed. Then there
is a section about the computation of the potential energy and dipole surfaces
on a grid, and how they are �tted with polynomials in AnharmND. The
eigenvalue equation is solved with a Lanczos scheme or a modi�ed Lanczos
scheme, which is described in the fourth section of this chapter. Finally in
the last section the expressions to compute integrated infrared absorption
intensities are derived from time dependent perturbation theory.
18 Chapter 2: Theory of the Computation of Infrared Spectra
2.1 The Hamiltonian and its Matrix Elements
2.1.1 The Vibrational Hamiltonian in Internal Linear Coordinates
Consider a system of N atoms with cartesian coordinates x1; : : : ; x3N , masses
m1; : : : ;mN , and M linear internal coordinates q1; : : : ; qM . In this paragraph
we will express the vibrational Hamiltonian in the coordinates fqig and their
conjugate momenta.
De�ne the matrix A such that
Aq = x: (1)
Note that because the coordinates fqig are linear, the elements of A are con-
stant. The Lagrangian L can be expressed in terms of the internal coordinates
fqig. First an expression for the kinetic energy T is derived:
T = 12
3NXi=1
mdi=3e _x2i
= 12
3NXi=1
mdi=3e
n MXj=1
Aij _qj
o2
= 12
MXj=1
MXk=1
n 3NXi=1
AijAikmdi=3e
o_qj _qk
= 12
MXj=1
MXk=1
Gjk _qj _qk;
(2)
where the matrix G is de�ned by
Gjk �3NXi=1
mdi=3eAijAik: (3)
The Lagrangian is de�ned as the di�erence between the kinetic energy T and
the potential energy V :
L = T � V = 12
MXj=1
MXk=1
Gjk _qj _qk � V (fqig): (4)
The Hamiltonian and its Matrix Elements 19
The momenta fpig conjugated to the coordinates fqig are found by di�eren-
tiation of the Lagrangian. Taking into account that G is symmetrical this
yields
pl =@L
@ _ql=
MXj=1
Gjl _qj ; (5)
so that we have
_qi =
MXj=1
(G�1)ijpj ; (6)
where (G�1)ij denotes an element of the inverse of G. Now the kinetic energy
can be written as a function of momenta:
T = 12
MXi=1
MXj=1
(G�1)ijpipj ; (7)
and the classical Hamiltonian becomes
H = T + V = 12
MXi=1
MXj=1
(G�1)ijpipj + V (fqig): (8)
Because the coordinates we are using are linear the matrices A and G are
constant, and we can convert the classical Hamiltonian Eq. 8 into a quan-
tum mechanical one by substitution of the momenta pj with the operators
(�h=i)=(d=dqj). If the coordinates used are not linear, the matrices A and G
become functions of fqjg, and the conversion of Eq. 8 into a quantum me-
chanical Hamiltonian becomes more complicated because qj and (�h=i)(d=dqj)
do not commute. The construction of a quantum mechanical Hamiltonian in
general coordinates has been described by Podolsky.[1]
20 Chapter 2: Theory of the Computation of Infrared Spectra
2.1.2 The Second Quantization Formalism for a Harmonic Oscillator
To facilitate calculations with vibrational wave functions, it can be advan-
tageous to switch from coordinates and conjugated momenta to a represen-
tation in creation and annihilation operators,[2] also known as the second
quantization formalism. These operators are de�ned in the context of a one-
dimensional harmonic oscillator. Consider a harmonic oscillator along coor-
dinate q with a reduced mass �, force constant � and conjugated momentum
p. The Hamiltonian for this oscillator is given by
H =p2
2�+ 1
2�q
2: (9)
The annihilator operator a and the creator operator ay are each other's Her-
mitian adjoints, de�ned as follows:
a ��!q + ip
p2��h!
ay �
�!q � ipp2��h!
:
(10)
where ! is de�ned asp�=�, �h is Planck's constant h divided by 2�, and
i2 = �1. Note that these operators are dimensionless by this de�nition. The
spatial coordinates and conjugated momenta can be expressed in creator and
annihilator operators:
q =
s�h
2�!(ay + a)
p = i
r�!�h
2(ay � a):
(11)
The quantum mechanical operators q and p do not commute, because in the
spatial coordinate representation p is given by p = (�h=i)(d=dq). From this the
commutator of q and p can be computed:
[q; p] � qp� pq =�h
i
(qd
dq
�d
dq
q) =�h
i
(qd
dq
� (1 + q
d
dq
)) = ��h
i
= i�h (12)
From the commutation relationship between a coordinate and its conjugated
moment, we �nd the commutation relation for creation and annihilation op-
erators.
[q; p] = i�h , [a; ay] = 1 (13)
The Hamiltonian and its Matrix Elements 21
The Hamiltonian Eq. 9 can now be rewritten in terms of creation and anni-
hilation operators:
H = 12�h!(aya+ aa
y) = �h!(aya+ 12): (14)
In Dirac notation, an eigenfunction of a harmonic oscillator is written as jni,where n is the excitation level or number of phonons or quanta in the state.
The e�ect of a and ay on a state jni is to remove or add a quantum, hence
their names.ajni =
pnjn� 1i
ayjni =
pn+ 1jn+ 1i
(15)
Matrix elements of combinations of annihilation and creation operators
are easily computed if they are put in the normal product form. A normal
product is de�ned by
n[�; �] � (ay)�(a)�: (16)
For multiplication we have the following relations (using Eq. 13):
Eq. 18 can be used recursively to express powers of q into normal products
of annihilation and creation operators. For the kinetic energy only p and p2,
given by Eq. 19, are of practical importance.
A simple analytical expression for matrix elements of normal products
can be deduced from Eq. 15:
hnjn[�; �]jmi =�q
n!m!(n��)!(m��)!
if n� � = m� � � 0;
0 otherwise.(20)
22 Chapter 2: Theory of the Computation of Infrared Spectra
2.1.3 A Basis Set for Multidimensional Vibrational Wave Functions
We approximate eigenfunctions of the Hamiltonian Eq. 8 with a variational
approach, expanding them in a basis set of products of Hermite functions of
each coordinate qi. Hermite functions are the eigenfunctions of a harmonic
oscillator. We need to be able to compute matrix elements of them, which
means we need to choose the � and ! parameters from Eq. 18 and Eq. 19. To
do this we split up the Hamiltonian in two parts, a `sum of harmonics' part
Hh and a `anharmonic plus coupling' part Ha+c:
H = Hh +Ha+c =
MXj=1
�12(G�1)jjp
2j +
12�jq
2j
+Ha+c: (21)
Ha+c contains the kinetic energy terms 12(G�1)jkpjpk and the bilinear po-
tential energy terms �jkqjqk with j 6= k, and the anharmonic terms in the
potential energy. In general it is best to choose the coordinates in such a way
that the o�-diagonal terms of the kinetic energy and the o�-diagonal bilinear
terms of the potential energy are zero. This means that the normal coordinates
for the used set of internal coordinates are used. There are occasions that this
may not be the best choice, e.g., in the description of a double well potential.
This chapter describes the method that is implemented in AnharmND, and
the program does not compute the normal coordinates by itself. This should
be done by the user.
The eigenfunctions � of Hh are of the form
�(n1; : : : ; nM ) =
MYj=1
�
(nj)j (qj) = jn1; : : : ; nmi; (22)
where �(nj)j (qj) is the eigenfunction of a harmonic oscillator in coordinate qj ,
containing n quanta. From comparison of Eq. 9 with Eq. 21 we determine the
parameters of these functions as
�j^
=1
(G�1)jjand !j =
q�j=�j
^
=
q�j(G�1)jj : (23)
The eigenfunctions of the complete Hamiltonian Eq. 8 are approximated by
a linear combination of the basis functions �. These basis functions form a
Fitting the Potential Energy and Dipole Surfaces 23
good basis set if the potential V has a minimum for q = 0. Eq. 18 and Eq. 19
now become
qj =
r12�h
q(G�1)jj=�j
�nj[1; 0] + nj[0; 1]
�(24a)
qj nj[�; �] =
r12�h
q(G�1)jj=�j
�nj[�+ 1; �] + nj[�; � + 1] + � nj[� � 1; �]
�(24b)
pj = i
r12�h
q�j=(G�1)jj
�nj[1; 0]� nj[0; 1]
�(25a)
p2j = �
12�h
q�j=(G�1)jj
�nj[2; 0]� 2 nj[1; 1]� nj[0; 0] + nj[0; 2]
�: (25b)
2.2 Fitting the Potential Energy and Dipole Surfaces
2.2.1 Considerations on Grids and Polynomials
AnharmND �ts potential energy and dipole surfaces speci�ed on a grid to
polynomials. The grid should cover the area where the wave functions of in-
terest and the used basis functions have non-negligible amplitude. To be able
to construct the grid this way, it is necessary to know approximately what the
fundamental frequencies of the modes being studied are. This information can
be obtained from a normal mode analysis with a standard quantum chemical
program like Gaussian or ADF, or by a numerical calculation of the force con-
stants. The harmonic force constants are the parameters that determine the
basis functions in AnharmND, together with the reduced masses. From them,
the r.m.s. (root-mean-square) width of the basis functions can be computed:
2ph�ijq2j�ii = 2
s(i+ 1
2)�h
p��
= 2
s(i+ 1
2)�h
2���: (26)
In this expression j�ii is the ith order Hermite function of coordinate q, � is
the reduced mass, � the force constant, � the frequency, �h Dirac's constant,
and � the ratio of the circumference of a circle and its diameter. It appears
that the grid needs to cover about two to four times the r.m.s. width of the
highest order basis function. No hard rules can be given.
The advantage of a polynomial as a functional form to �t a potential or
dipole component, is that it has no bias to a particular shape. Most potential
wells are well suited for �tting with a polynomial. The biggest disadvantage
in AnharmND is the behaviour of polynomials outside the area of the grid
points: they go either to +1 or to �1. If the polynomial describing the
24 Chapter 2: Theory of the Computation of Infrared Spectra
potential goes to �1 for certain coordinate values this poses problems if
there are basis functions that have non-negligible amplitude there. Unphysical
wave functions with low energy will be calculated that can be recognized by
high order Hermite functions in the basis functions with large coeÆcients.
If it occurs, increasing the basis set will give a continuously lower ground
level involving the highest order Hermite functions from the basis set. To
prevent this situation, we start by only using polynomials of which the highest
order terms are of even order, and try to choose the grid in such a way that
the coeÆcients of those terms are positive. If this is not possible, the basis
functions must be chosen in such a way that they have no large amplitude in
the area where the �tted potential has unphysically low values.
The number of points in a grid is a compromise between accuracy and
computer time. To be able to include anharmonic e�ects, and to �t with an
even order polynomial, the smallest number of points that can be used is �ve
along each mode, with a fourth order polynomial. Though this seems a small
number, it is in fact quite large once multiple dimensions come into play. If
�ve points along each mode are used in four dimensions and a regular grid
is constructed, the grid contains 54 = 625 points. The next logical grid size
would be 74 = 2401, which explains why in this thesis �ve points along each
mode has been deemed plenty. It should be noted that the number of grid
points can be lowered leaving out the points of the grid where more than
one coordinate has an extreme value, since these points tend to have such
high energy that they are unimportant for the wave functions of lower energy.
Still, the general trend is that the number of points in the grid need to �t a
polynomial of increases with the exponent of the order of the polynomial.
The energy polynomials preferably have the following shape:
E(q1; : : : ; qD) =X
�1;:::;�D
c�1;:::;�D
DYi=1
q�ii with 0 �
DXi=1
�i � N: (27)
Here D is the number of dimensions or coordinates qi, N is the order of the
polynomial, and c�1;:::;�D are coeÆcients for the products of powers �i of the
coordinates. If the condition on the right of Eq. 27 is maintained, the poly-
nomial can remain invariant under linear transformations of the coordinates
qi.
Weight factors have been used in the �t to assign greater weight to grid
points with lower energy, because we want to start with a good description
of the ground state, and are only interested in a relative small number of
excitations per mode. The weight factors are calculated proportional to the
negative exponent of the energy times an `energy scale factor', and normalized
so that the r.m.s. error of the �t can be interpreted in a meaningful way. The
energy scale factor used throughout this thesis is 125Eh�1.
Fitting the Potential Energy and Dipole Surfaces 25
2.2.2 Performing a Least Squares Fit with Singular Value Decomposition
Given the energy polynomial of Eq. 27, we have an equation for each energy
Ej calculated on the grid to �t the polynomial to:
Ej(q1j ; : : : ; qDj) =X
�1;:::;�D
c�1;:::;�D
DYi=1
q�iij ; j = 1; : : : ;M: (28)
Here qij is the value of coordinate qi for point j of the grid. The coeÆcients
c�1;:::;�D are the unknowns in these equations. If we suppose there are N
coeÆcients, and M grid points, we can write the equations in matrix form.
Qc = e (29)
Here Q is theM�N matrix containing the products of powers of coordinates
qi, c is the vector containing the coeÆcients c�1;:::;�D , and e is the vector
containing the energies from the grid points. The problem at hand is to �nd
a vector c such that r � jQc� ej is minimal, where r is called the `residual'.
This is the least squares approximation of the coeÆcients. If the number of
equations M is equal to the number of coeÆcients N , there exists a unique
solution with r = 0 to Eq. 29, provided there are no linear dependencies
between the equations. In practice, M and N are usually not equal and
you do not know in advance if the �t equations contain linear dependencies.
The method to tackle the situation is singular value decomposition. [3, and
references therein]
In linear algebra there exists a theorem that any M �N matrix A, with
N �M , can be decomposed into a column-orthonormal matrix U, a diagonal
matrix W, and the transpose of an orthonormal matrix V:
0BBBBB@
a11 � � � a1N...
...
ai1 � � � aiN...
...
aM1 � � � aMN
1CCCCCA =
0BBBBB@
u11 � � � u1N...
...
ui1 � � � uiN...
...
uM1 � � � uMN
1CCCCCA0@w1
. . .
wN
1A0@ v11 � � � vN1
.... . .
...
v1N � � � vNN
1A: (30)
26 Chapter 2: Theory of the Computation of Infrared Spectra
This decomposition can always be done, and is unique except for two trans-
formations: �rstly permutations of the columns of U, the elements ofW, and
the rows of VT ; secondly linear combinations of columns of U, and of rows of
VT whose corresponding elements of W are exactly equal.
If A is square and all elements of W are non-zero, it becomes easy to
compute the inverse of A with the singular value decomposition:
A�1 = V � [diag(1=wi)] �UT: (31)
This decomposition also shows when it becomes impossible to compute the
inverse of A, namely, if one of the singular values wi becomes zero. The con-
dition number of A is the ratio between the largest and the smallest element
of W. If this number becomes large, A becomes ill-conditioned. A condition
number is large if its reciprocal value is in the neighbourhood of the precision
of the computer you are working with, but there are more criteria. If the
condition number becomes in�nite, the matrix is singular.
SVD helps to �nd a useful answer to the equation Ax = b, even if A
is singular or ill-conditioned. The columns of U that correspond to non-zero
elements of W represent the range of A, whereas the columns of V that
correspond to elements of W that are zero span the null-space of A. If b lies
in the range of A, then the equations can be solved with SVD, replacing 1=wiwith 0 if wi = 0:
Ax = b ) x = V � [diag(1=wj)] � (UT � b): (32)
This yields the vector x with smallest length, and any linear combination of
the vectors from the null-space of A can be added to it. If b does not lie in
the range of A, then the equation cannot be solved, but x is the solution in
the least squares sense: the residual is minimized.
In practice, if there are elements of W that are very small compared
to the largest wi, 1=wi is also set to zero when computing x like in Eq. 32.
The reason is that these elements correspond to equations in Ax = b that
make large di�erences in x (1=wi is large), but contribute very little to the
improvement of the �t (i.e., the diminishing of the residual). The reason that
this is so can be numerical, but also inherent to the physical data. The best
way to get an idea for the limit below which to set 1=wi to zero is to order
the wi in decreasing size. Usually there is a gap of a few orders of magnitude
between the wi that are `OK', and those that should be zeroed. Like in the
choice of a grid and a basis set, there is a certain amount of Fingerspitzengef�uhl
involved.
Eq. 32, including zeroed 1=wi values, also applies to non-square matrices
A, where the number of equationsM is greater than the number of coeÆcients
Solving the Eigenvalue Problem with the Lanczos Method 27
N . In the case there are less equations than coeÆcients, the matrix A and
the vector b can be �lled up with zeroes until A is square. If this is done, or
indeed if any of the 1=wi need to be zeroed, the equations do not completely
determine the coeÆcients. In the application of SVD to the �t of a polynomial
to the potential energy grid, the �tted polynomial can still be used, and the
way it is obtained seems less arbitrary than leaving out certain coeÆcients
from the �t altogether.
SVD applied to Eq. 29 �nds the coeÆcients that minimize jQc�ej, whichcorresponds to an unweighted least squares �t. As mentioned earlier, the �ts
that are used in AnharmND are weighted. To get a properly weighted �t, we
apply SVD on
h[diag(
pw0
i)] �Qi� c =
h[diag(
pw0
i)] � ei; (33)
where w0
i are the �t weights (not to be confused with the singular values wi)
belonging to the energies Ei. The �t weights are calculated as
w0
i =e�f �EiPM
j=1 e�f �Ej
; (34)
where f is the energy scale factor mentioned earlier. For �t weights de�ned
in this way, the residual of Eq. 33 is the properly weighted r.m.s. error of the
�t on a grid point.
In AnharmND, the x, y, and z-component of the dipole are �tted to
polynomials as well, and the same weights are applied as for the �t of the
energy. However, it is possible to specify the dipole components for less grid
points than the energy, and then the weight factors are scaled so that the
r.m.s. error is still calculated per grid point where a dipole was given.
2.3 Solving the Eigenvalue Problem with the Lanczos Method
For the solution of the Schr�odinger equation, the lowest eigenvalues and eigen-
vectors of the Hamiltonian need to be computed. The Hamiltonian both sparse
and symmetric. Because of these properties, it is well suited to be tackled with
the Lanczos method.
28 Chapter 2: Theory of the Computation of Infrared Spectra
2.3.1 The Exact Lanczos Method
In its initial incarnation, the Lanczos method is a method to transform a
symmetrical matrix A of size n � n into a tridiagonal form T, with a trans-
formation matrix Y � [y1 y2 : : : yn], where fyig form an orthonormal set of
vectors. The eigenvalues and vectors of a tridiagonal matrix can be computed
readily.
AY = YT = [y1 y2 : : : yn]
266666664
�1 �2
�2 �2 �3
�3 �3 �4
�4
. . .. . .
. . .. . . �n
�n �n
377777775
(35)
Paige [4] found two numerically stable algorithms to compute the vectors
yi and the tridiagonal matrix T to �nd eigenvalues. We use the following
algorithm. Take a random vector y1 of unit length, let �1 = 0, and for i = 1
to n, do
1. v Ayi.
2. �i yTi v.
3. z v � �iyi � �iyi�1.4. �i+1
pzTz.
5. if �i+1 = 0, then stop; otherwise, yi+1 z=�i+1 and continue.
(36)
It can be shown that, for exact arithmetic, the vectors yi form an orthonormal
set. In exact arithmetic, the algorithm stops if �i+1 equals 0. This can happen
at �i+1 if y1 is orthogonal to n � i eigenvalues of A. If �i+1 = 0 and i < n
then Yi = [y1 y2 : : : yi] is an invariant subspace, and the algorithm can
be continued with a vector yi+1 that is orthogonal to all y1; : : : ;yi. The
algorithm has to end with �n+1 = 0 because you cannot have more than n
orthogonal vectors of size n.
After the construction of the tridiagonal matrix T, its eigenvalues �iwith eigenvectors ti can be computed. The eigenvalues �i of the symmetrical
matrixA are identical to those of T, and the eigenvectors ai can be calculated
if the Lanczos vectors yi have been stored, by ai = Yti.
Solving the Eigenvalue Problem with the Lanczos Method 29
2.3.2 The Blessing of Round-O� Errors in Lanczos
In exact arithmetic the Lanczos vectors that are generated in the Lanczos
procedure are all orthogonal, but in �nite accuracy calculations such as they
occur in computers, they are not. In the procedure outlined in the previ-
ous paragraph, yi is orthogonalised explicitly only to yi�1 and to yi�2. The
implicit orthogonality with respect to the other yi is gradually lost after a
number of iterations. As a result the matrix T will not have exactly the same
eigenvalues as A, and the eigenvectors of A are not exactly reproduced by
Yti. Originally, Lanczos tried to �x the method by reorthogonalisation of the
vectors yi against all previous ones. This works, but the associated computa-
tional cost is prohibitive for large matrices A, both in terms of memory and
CPU usage.
Paige showed that for large numbers of iterations, a good number of the
extreme eigenvalues of A can be found. If the number of iterations exceeds
the dimension n of A, a few conceptual problems appear. They are related to
the fact that YTi AYi has a greater dimension than A. This means that more
eigenvalues can be computed from T than A actually has, some of which are
multiple copies of eigenvalues that A has, others which are not eigenvalues
of A at all. These super uous eigenvalues are called `spurious multiplicities'
and `spurious eigenvalues' respectively, and a practical implementation of the
Lanczos method needs a test to distinguish them from `genuine' eigenvalues.
The test we applied will be discussed further on. Spurious multiplicities and
eigenvalues already appear in the case that the number of iterations does not
exceed the dimension of A.
Although it seems an annoying complication at �rst sight, the loss of
orthogonality in the Lanczos method with �nite accuracy is the very reason
it works in practice. The Lanczos method only �nds eigenvectors that are
in the part of the eigenspace spanned by Yi. In exact arithmetic this means
that y1 must be a linear combination of all eigenvectors of A to continue the
iteration to Yn without getting �i+1 = 0 before the complete vector space
is spanned. This is a tough call on a randomly chosen vector. In the �nite
accuracy practice of computers the event that �i+1 becomes zero is extremely
rare. If �i+1 in the procedure given in the previous section becomes small,
rounding errors in z in step 5 are transferred much in ated into yi+1. This
is where loss of orthogonality is introduced, but with it may also come parts
of the eigenvector space not yet present in y1. If enough Lanczos iterations
are carried out, a good number of extreme eigenvalues are found, even if
the corresponding eigenvectors ai are orthogonal to the �rst random Lanczos
vector.
30 Chapter 2: Theory of the Computation of Infrared Spectra
If loss of orthogonality is severe, the correspondence between the eigen-
values of T and A will be poor whereas if the columns of Y are perfectly or-
thogonal, there is the risk of missing out part of the eigenspace of A. Lewis [5]
recommends partial reorthogonalisation of vectors yi against yi�1 and yi�1.
This appears to provide enough orthogonality to obtain more eigenvalues than
just the most extreme ones. The procedure then becomes
1. v Ayi.
2. �i yTi v.
3. z v � �iyi � �iyi�1.4. z z� (zTyi)yi:
5. z z� (zTyi�1)yi�1:
6. �i+1 pzTz.
7. if �i+1 = 0, then stop; otherwise, yi+1 z=�i+1 and continue.
(37)
In the computer version of the algorithm, if �i+1 = 0, the procedure is sim-
ply restarted with a di�erent random starting vector. In practice this never
happens.
2.3.3 Getting Rid of Spurious Eigenpairs
We compute infrared absorption intensities and expectation values of coordi-
nates. Both properties can be computed straightforwardly using the eigen-
vectors of the Hamiltonian. If the eigenvectors are going to be computed
anyway, there is a convenient way to compute the accuracy of the eigenvalues.
In Dirac notation, if there is a normalized vector j i with an expectation valueh jHj i, then there exists an eigenvalue � of the operator H such that [6]
jh jHj i � �j �ph jH2j i � jh jHj ij2: (38)
This expression provides a way to discover spurious eigenvalues: if the in-
terval computed for a vector j i is too wide, it is not a good approximation
of an eigenvector. Spurious multiplicities are found by orthogonalisation of
eigenvectors of the tridiagonal matrix that have overlapping intervals contain-
ing eigenvalues of the Hamiltonian. If the orthogonalisation makes a vector
vanish, it counts as a spurious multiplicity.
Solving the Eigenvalue Problem with the Lanczos Method 31
2.3.4 A Modi�ed Lanczos Procedure
For some of the infrared spectra we have computed, we need so many eigen-
values and vectors that the number of Lanczos iterations necessary to �nd
them becomes prohibitive in terms of computation time and use of memory,
especially because we want to �nd the eigenvectors as well as the eigenval-
ues. There are two properties that make an eigenvector likely to be found in
a Lanczos procedure: being at the extremes of the eigenspectrum, and be-
ing well separated from other eigenvalues. We used a method that aims at
improving the separation of the eigenvalues we search, at the expense of the
eigenvalues we are not interested in. In this procedure which is described by
Lewis [5], the original Hamiltonian H is replaced with a polynomial of this
Hamiltonian f(H). The function f modi�es the eigenspectrum of H, whereas
the eigenvectors of f(H) are the same as those of H.
For the construction of f we �rst determine the extreme eigenvalues of
the Hamiltonian H. This can be done easily with a standard Lanczos pro-
cedure. Once we know the interval [Emin; Emax] in which the eigenvalues lie,
we choose a value E0 2 [Emin; Emax] that is an upper limit for eigenvalues
we want to compute. Then we construct a polynomial that `folds up' the
interval [E0; Emax], and stretches the interval [Emin; E0], such as fourth order
polynomial in the following graph:
fmax
f0
fmin
Emin E0 Emax
The interval [f(Emin); f(E0)] has much better separated eigenvalues compared
to [f(E0); f(Emax)], than the interval [Emin; E0] has compared to [E0; Emax].
Eigenvalues of H can be computed as expectation values of the eigenvectors
found for f(H), and tested for accuracy in the same way as in the plain
Lanczos procedure. Using this modi�cation it appears possible to obtain
higher eigenvalues in less iterations.
32 Chapter 2: Theory of the Computation of Infrared Spectra
2.4 Infrared Absorption Intensities
2.4.1 The Law of Lambert and Beer, Integrated Absorption Intensities
The law of Lambert and Beer describes the absorption of radiative energy in
a medium:
I(�) = I0 e�k(�)�x
: (39)
I0 in this formula denotes the initial intensity, I(�) the intensity at frequency
�, k(�) the absorption coeÆcient for frequency (�), and x is a quantity denot-
ing the number of absorbing molecules per unit surface area. The quantity x
has been de�ned in disturbingly many di�erent ways.[7] It is the product of a
concentration and a length, and for both quantities di�erent units are being
used. AnharmND uses
x � n � l; (40)
where n is the concentration in mol/m3, and l is a length the radiation travels
through, in m. The absorption coeÆcient k(�) can be written as follows:
k(�) = S f(� � �0): (41)
Here S is the integrated absorption coeÆcient, also called line strength or
line intensity, and f(� � �0) is a function describing the line shape, that is
normalised by the requirement
Z1
�1
f(� � �0)d� = 1: (42)
From Eq. 41 and 42, we �nd that
S =
Z1
�1
k(�)d�: (43)
This expression shows us that the units in which S is expressed are the units of
� divided by the units of x. AnharmND uses the same convention as quantum
chemical programs such as Gaussian, where � is in wave numbers. As a result,
S is expressed in km=mol.
Infrared Absorption Intensities 33
2.4.2 Einstein CoeÆcients
Einstein has established the relationship between absorption and emission of
radiation of a system with two states in a radiation bath in thermal equilib-
rium.[8] From his analysis we get the de�nitions of the so-called Einstein A
and B coeÆcients.
Assume a system that has two stationary states with energiesEm and En,
where Em > En. Absorption of radiation with frequency �nm will bring the
system from state n to m. The frequency �nm follows from Bohr's frequency
rule:
�nm =Em �En
h
: (44)
The radiative density is given by �(�), where �(�)d� is the energy of radiation
with frequency between � and � + d�. The probability of the system in state
n to absorb a quantum in a unit of time is given by
Bn!m�(�nm); (45)
where Bn!m is the Einstein coeÆcient of absorption. Einstein has shown that
the probability of emission should be divided into two parts: spontaneous and
stimulated emission. The probability for the system in state m to emit a
quantum per unit time is given by
Am!n +Bm!n�(�nm); (46)
where Am!n is Einstein's coeÆcient of spontaneous emission, and Bm!n is
the Einstein coeÆcient of stimulated emission.
Consider a large number of systems in equilibrium with radiation at tem-
perature T . According to Planck,[8] the distribution of radiation is given
by
�(�) =8�h�3
c3
(eh�=kT � 1)�1: (47)
Assume Nn systems in state n, and Nm systems in state m. At equilibrium
the number of transitions from n to m, NnBn!m�(�nm), and the number
of transitions from m to n, Nm[Am!n +Bm!n�(�nm)], are equal. From this
equality an expression can be derived for the ratio between Nn and Nm, which
can then be equated to the ratio according to the Boltzmann distribution:
Nn
Nm
=Am!n + Bm!n�(�nm)
Bn!m�(�nm)= e
�(En�Em)=kT = eh�nm=kT
: (48)
This expression can be rewritten as
�(�nm) =Am!n
Bn!meh�nm=kT �Bm!n
: (49)
34 Chapter 2: Theory of the Computation of Infrared Spectra
To match this with Planck's law, the following two requirements arise:
Bn!m = Bm!n
Am!n =8�h�3nmc3
Bm!n:
(50)
2.4.3 Time Dependent Perturbation Theory
To compute the transition probability for a system from one state to another
in quantum mechanics, we apply time dependent perturbation theory.[9] The
time dependent Schr�odinger equation is given by
Hji = i�h@
@t
ji: (51)
If the Hamiltonian H does not depend on time, it can be solved by separation
of variables:
ji = j ie�iEt=�h; (52)
where j i is the solution to the time-independent Schr�odinger equation
Hj i = Ej i: (53)
Consider a system with orthonormal states j ji that are eigenstates of the
time-independent Hamiltonian H0. Any state of the system can be expressed
as a linear combination of these states, e.g.
jit=0 =Xj
cj j ji (54)
or, more general,
ji =Xj
cj j jie�iEjt=�h (55)
Normalization of ji requires thatXj
jcj j2 = 1: (56)
Since the states j ji form a complete set, Eq. 55 still holds when a time de-
pendent perturbationH 0 is added to the Hamiltonian, with the di�erence that
the formerly constant coeÆcients cj now become functions of time. For this
reason time-dependent perturbation theory is sometimes called the `method
of variation of constants'. If we start with cn(0) = 1 and cm(0) = 0 for m 6= n,
Infrared Absorption Intensities 35
and at some later time t1 we have cn(t1) = 0 and jcm(t1)j2 = 1, we say that
the system underwent a transition from j ni to j mi.To �nd cn(t) and cm(t), we require that ji from Eq. 55 satisfy the time
dependent Schr�odinger equation with Hamiltonian H0 +H0.
Xj
cj [H0j ji]e�iEjt=�h +Xj
cj [H0j ji]e�iEjt=�h =
Xj
i�h _cj j jie�iEjt=�h +Xj
i�hcj@
@t
�j jie�iEjt=�h
� (57)
The �rst and the last sum in this equation cancel because j ji are eigenfunc-tions of H0, so that we getX
j
cj [H0j ji]e�iEjt=�h =
Xj
i�h _cj j jie�iEjt=�h (58)
Separate expressions for each _ck can be obtained by taking the inner product
of Eq. 58 with h kj
Xj
cjh kjH 0j jie�iEjt=�h = i�h _cke�iEkt=�h (59)
We de�ne the following shorthand:
H0
ab � h ajH0j bi (60)
Eq. 59 can be rewritten as
_ck = �i
�h
hckH
0
kk +Xj 6=k
cjH0
kje�i(Ej�Ek)t=�h
i: (61)
Typically the diagonal matrix elements of H 0 are zero, which gives the follow-
ing equation for each _ck:
_ck = �i
�h
Xj 6=k
H0
kje�2�i�kjt
cj ; (62)
where �kj is de�ned as in Eq. 44.
In perturbation theory, equations 62 are solved by successive approxi-
mations, assuming that H 0 is small. Consider the system starts in the lower
36 Chapter 2: Theory of the Computation of Infrared Spectra
state j ni. In the absence of a perturbation, the system would stay like this
forever. This is the zeroth order approximation:
c(0)n (t) = 1; c
(0)
m6=n(t) = 0: (63)
We get the �rst order approximation inserting these values in Eq. 62 (taking
k = n, and k = m, respectively):
dcn
dt
= 0 ) c(1)n (t) = 1
dcm
dt
= �i
�hH
0
mne2�i�nmt ) c
(1)m (t) = �
i
�h
Z t
0
H0
mn(t0)e2�i�nmt
0
dt0
(64)
Obviously, c(1)n (t) and c
(1)m (t) do not ful�ll the normalization condition Eq. 56.
This is because they are only correct to �rst order. The second order ap-
proximation can be obtained inserting c(1)n (t) and c
(1)m (t) into the right hand
side of Eq. 62. This will give a new expression for c(2)n (t), but the expression
for c(2)m (t) will be identical to the one for c
(1)m (t). Since we are interested in
the development of cm(t), the �rst order approximation gives a useful result
already, that is in fact correct to second order.
2.4.4 Quantum Mechanical Calculation of the Einstein CoeÆcients
With the aid of time dependent perturbation theory, and a classical treatment
of the electromagnetic �eld, an expression can be derived for the Einstein
coeÆcients.
In classical electrodynamics, an electromagnetic wave consist of trans-
verse and mutually perpendicular electric and magnetic �elds. Atoms and
molecules interact primarily with the electric component of such waves. For
visible and infrared light the wave length is several orders of magnitude larger
than the molecular dimensions, so that the electric �eld can be considered
constant in space, but varying in time:
E = E0 cos(2��t)k: (65)
Here E is the electric �eld with frequency �, amplitude E0, and polarisation
vector k of unit length. If we assume that E is polarised in the z-direction,
the perturbing Hamiltonian H 0 is given by
H0 =
Xj
QjE0zj cos(2��t) = �zE0 cos(2��t); (66)
Infrared Absorption Intensities 37
where Qj are the charges in the system with z-coordinate zj , and �z is the
z-component of the dipole moment.
Before we proceed inserting this expression in Eq. 64, we need to re-
address the diagonal elements H 0
kk in Eq. 59 and 61. If the described system
is an atom, the wave functions j ki are usually either even or odd functions
of z, and �z is an odd function of z. This causes diagonal matrix elements
of H 0 to vanish. In this thesis j ki usually represents a vibrational wave
function that often is not purely even or odd, and �z often contains a constant
term. If the diagonal terms are not set to zero, this changes the �rst order
approximation of cn(t), but the c(1)m (t) remains the same as in Eq. 64. Since
we are interested in cm(t), we can continue to use Eq. 64 as the �rst order
approximation of cm(t), but it can no longer be regarded as correct to second
order for all vibrational modes.
A matrix element of H 0 can be split in a time dependent and a time
independent part:
H0
ab = h aj�zE0 cos(2��t)j bi = E0h aj�zj bi cos(2��t) = Vab cos(2��t);
where Vab � E0h aj�zj bi:(67)
We can insert this into the expression for c(1)m (t) from Eq. 64, to get
cm(t) � �iVmn
�h
Z t
0
cos(2��t0)e2�i�nmt0
dt0
= �iVmn
2�h
Z t
0
he2�i(�nm+�)t0 + e
2�i(�nm��)t0
idt0
= �Vmn
4��h
�e2�i(�nm+�)t � 1
�nm + �
+e2�i(�nm��)t � 1
�nm � �
�:
(68)
This expression can be simpli�ed because we are mainly interested in fre-
quencies � in the neighbourhood of �nm, and frequencies far from that value
are not likely to cause a transition anyway. This means that the left term be-
tween the brackets in Eq. 68 becomes negligible compared to the right term,
and it can be dropped:
cm(t) � �Vmn
4��h
�e�i(�nm��)t � e��i(�nm��)t
�nm � �
�e�i(�nm��)t
= �iVmn
2��h
sin(�(�nm � �)t)�nm � �
e�i(�nm��)t
:
(69)
38 Chapter 2: Theory of the Computation of Infrared Spectra
From this we �nd that the probability Pn!m to �nd the system in j mi attime t, if it was in j ni at time t = 0, under the in uence of an electric �eld
polarised in the z direction, is given by
Pn!m(t) = jcm(t)j2 �jVmnj2
4�2�h2sin2[�(�nm � �)t]
(�nm � �)2
=E20 jh mj�z j nij2
4�2�h2sin2[�(�nm � �)t]
(�nm � �)2:
(70)
In classical electrodynamics, the energy per unit volume u in electromag-
netic �elds is given by
u =�0
2E2 +
1
2�0B2; (71)
where E and B are the electric and magnetic �elds, �0 is the electric per-
mittivity of vacuum, and �0 is the magnetic permeability of vacuum. For
electromagnetic waves the electric and magnetic contributions are equal. On
the average over a cycle the energy in an electromagnetic wave is then given
by
u =�0
2E20 : (72)
This means that the transition probability from Eq. 70 is proportional to the
energy density of the �elds:
Pn!m(t) =ujh mj�z j nij2
2�2�0�h2
sin2[�(�nm � �)t](�nm � �)2
: (73)
This holds for a monochromatic perturbation, of frequency �. In realistic
situations the system is exposed to electromagnetic waves in a range of fre-
quencies. This means that we need to replace u with an energy density �(�)d�
in the range d�, and integrate over �:
Pn!m(t) =jh mj�zj nij2
2�2�0�h2
Z1
0
�(�)
�sin2[�(�nm � �)t]
(�nm � �)2
�d�: (74)
The term in curly braces in Eq. 74 has a sharp peak around � = �nm, whereas
�(�) in general is a much broader function (Eq. 47), so we can approximate
�(�) with �(�nm):
Pn!m(t) �jh mj�zj nij2
2�2�0�h2
�(�nm)
Z1
0
�sin2[�(�nm � �)t]
(�nm � �)2
�d�: (75)
Infrared Absorption Intensities 39
The integration range in Eq. 75 can be extended to �1 : : :1, because the
function only has non-negligible value around � = �nm. Then after substitu-
tion of x � �(�nm � �)t, and using
Z1
�1
sin2 x
x2dx = �; (76)
we get
Pn!m(t) �jh mj�z j nij2
2�0�h2
�(�nm)t: (77)
From this we can calculate the transition rate Rz for polarised light along the
z-axis as
Rzn!m =
dPn!m
dt
=jh mj�zj nij2
2�0�h2
�(�nm): (78)
This expression holds if the dipole moment of the system and the polarisation
of the electric �eld are both in the same direction. For the general case we need
to replace jVnmj2 in Eq. 70 by E20 jh mj�j nij2 � (k �m)2 where the orientation
factor (k �m)2 is the square of the inner product of the polarisation direction
k of the electric �eld with the direction of the dipole moment m, averaged
over all possible relative orientations. Both k and m have unit length. The
orientation factor can be computed by
(k �m)2 =
R 2�0
d�
R �0d� cos2 � sin �R 2�
0d�
R �0d� sin �
=1
3: (79)
This expression is found taking � as the angle between k and m, and � the
rotation angle of m around k at �xed �. For given values of � and � the
square of the inner product is then cos2 �. With this result we can rewrite
Eq. 78 for the transition rate Rn!m for the system in any orientation with
electromagnetic radiation coming from all directions:
Rn!m =jh mj�j nij2
6�0�h2
�(�nm): (80)
By comparison of this expression with Eq. 45 we �nd the expression for the
Einstein coeÆcient of absorption:
Bn!m =jh mj�j nij2
6�0�h2
: (81)
40 Chapter 2: Theory of the Computation of Infrared Spectra
2.4.5 Calculation of the Integrated Absorption CoeÆcient
We need to make a connection between Einstein CoeÆcients and measurable
quantities.[10] Assume a beam of radiation with intensity I shines on a con-
tainer with our system with two states j ni and j mi. The di�erential changein intensity can be given by
�dI = h�nmBn!m�(�nm)Nndl; (82)
where Bn!m�(�nm) is the number of transitions per second of one system in
the presence of radiation energy density �(�nm) (see Eq. 45), h�nm the energy
absorbed by one transition, Nn the number of particles in state j ni per unitvolume, and dl the distance traveled through the sample.
The relationship between the energy ux or intensity I and the radiation
energy density �(�) is given by
I =
Zc�(�)d�; (83)
where c is the speed of light. For the frequencies around �nm that are of im-
portance for the transition we are studying, �(�) can be regarded as constant
(compare Eq. 75). This means that in that frequency range I(�) can also be
regarded as constant, and we get
I(�nm) = c�(�nm)��; (84)
where �� is the width of the peak. This can be substituted in Eq. 82, to give
�dI =h�nm
c��Bn!mI(�nm)Nndl: (85)
Compare this to the di�erential form of Eq. 39:
�dI = k(�)I(�)ndl: (86)
This gives
k(�)�� �NAh�nm
c
Bn!m (87)
where Avogadro's number NA appears because the units of n are mol/m3 and
those of Nn are molecules per m3. This equation is approximate because k(�)
is not constant over the width of the peak. Since in the calculation of the Ein-
stein coeÆcient the frequency dependency has been integrated out (Eq. 75), it
follows that the same should be done with the absorption coeÆcient, so thatZk(�)d� =
NAh�nm
c
Bn!m: (88)
Infrared Absorption Intensities 41
In practice, the integration is usually not carried out over frequency �, but over
wave numbers �=c. This means that for the integrated absorption coeÆcient
S in m/mol we have the following quantum mechanical expression:
Sn!m =
Zk(�)
c
d� =NAh�nm
c2
Bn!m =2�2NA�E
3�0h2c2jh mj�j nij2; (89)
where �E is the energy di�erence between j mi and j ni. For absorption
spectra of multiple levels at higher temperatures than 0K, a Boltzmann dis-
tribution factor is added to account for occupation of levels higher than the
ground state:
S(T )n!m =2�2NA�E
3�0h2c2jh mj�j nij2
e�En=kTPMj e
�Ej=kT: (90)
In this expression M is the number of states, and Ej is the energy of j ji.Eq. 90 has been used throughout this thesis in the computation of infrared
spectra. It disregards the e�ects of stimulated emission, that are proportional
to the occupation of the excited level involved in a transition. The e�ect of
induced emission can be accounted for subtracting a term from the Boltzmann
distribution factor, so that it becomes
e�En=kT � e�Em=kTPM
j e�Ej=kT
: (91)
This means that all intensities in the spectra could be corrected if they were
multiplied by the factor
e�En=kT � e�Em=kT
e�En=kT
= 1� e��E=kT (92)
At a temperature of 298.15K this factor as a function of wave numbers looks
like this:
0
0.99
0 2000
inte
nsity
rat
io
wave number (1/cm)
Influence of Induced Emission on Intensity at 298.15K
144 477 954
0.50
0.90
42 Chapter 2: Theory of the Computation of Infrared Spectra
The correction diminishes the intensity by a factor of 0.5 at 144 cm�1, 0.9
at 477 cm�1, and 0.99 at 954 cm�1. The most interesting transitions of the
spectra in this thesis are in the region where the e�ect of induced emission
can be neglected.
References
[1] B. Podolsky; Phys. Rev., \Quantum-Mechanically Correct Form of Ha-
miltonian Function for Conservative Systems", 32, 812{816 (1928).
[2] A. Messiah; Quantum Mechanics; chap. XII, North Holland, Amsterdam,
1962.
[3] W. H. Press, S. A. Teukolsky, and W. T. Vetterling; Numerical recipes in
C : the art of scienti�c computing; chap. 2.9, 14.3, Cambridge University
Press, Cambridge, 1992.
[4] C. C. Paige; J. Inst.Maths. Appl., \Computational Variants of the Lanc-
zos Method for the Eigenproblem", 10, 373{381 (1972).
[5] J. G. Lewis; Algorithms for sparse matrix eigenvalue problems; Stanford
University, Department Computer Science, Stanford, 1977.
[6] B. N. Parlett; The Symmetric Eigenvalue Problem; Prentice-Hall, Engle-
wood Cli�s, 1980.
[7] K. Narahari Rao; (editor)Molecular Spectroscopy: Modern Research, vol.
2; chap. 4 by L. A. Pugh, and K. Nahari Rao; \Intensities from Infrared
Spectra", Academic Press, London, 1976.
[8] L. Pauling, and E. B. Wilson Jr.; Introduction to quantum mechanics
with applications to chemistry; chap. XI-40a, McGraw-Hill, London, 1935.
[9] D. J. GriÆths; Introduction to Quantum Mechanics; chap. 9, Prentice
Hall, Englewood Cli�s, 1995.
[10] E. B. Wilson Jr., J. C. Decius, and P. C. Cross; Molecular Vibra-
tions: The Theory of Infrared and Raman Vibrational Spectra; chap. 7-9,
McGraw-Hill, London, 1955.
3Three and Four-Coordinate Cluster Models
Abstract
The in uence of acetonitrile adsorption on the infrared spectrum of an acidic
OH group inside a zeolite is studied by theoretical calculations. The zeolite
is modelled by a cluster molecule. Potential energy and dipole surfaces of
the stretch and two bending coordinates of the acidic H atom, and, for the
complex with acetonitrile, of an additional acetonitrile stretch coordinate,
are computed employing Hartree-Fock as well as density functional methods.
Infrared frequencies as well as absorption intensities are computed taking into
account mechanical as well as electric anharmonicities up to fourth order.
Fermi resonance proposed as cause of the splitting of OH stretch absorption
bands in infrared spectra is explicitly considered.
Reproduced in part with permission from E.L. Meijer, R.A. van Santen, and
A.P.J. Jansen: \Computation of the Infrared Spectrum of an Acidic Zeolite
Proton Interacting with Acetonitrile", J. Phys. Chem. 100 9282{9291 (1996).
Copyright 1996 American Chemical Society.
44 Chapter 3: Three and Four-Coordinate Cluster Models
3.1 Introduction
The infrared spectrum of an XH group in a molecule, where X is typically one
of O, S, F, Cl, Br, or I, can change radically upon hydrogen bonding to some
base B. In many di�erent systems the following changes are observed. The
XH stretching frequency is lowered considerably, the absorption intensity of
the band is enhanced greatly, and the band is broadened by a large amount.
In a number of cases the hydrogen bonded XH stretch band is split into two
or more bands. All of the described phenomena appear to be stronger if the
acidity of the XH group is greater, or if the basic character of B is stronger.
The system under study in the present work is a Br�nsted acidic OH
group of a zeolite interacting with a molecule of acetonitrile. This system is
especially interesting because it exhibits a very large OH stretch frequency
shift, and also the splitting of the shifted OH stretch band. In the case of
adsorption of methanol in a zeolite, a similar multiple band structure has
been found. Kubelkov�a et al.[1] have argued the two bands around 2400 cm�1
and 2900 cm�1 were due to two di�erent structures, in one of which the proton
is transferred to the methanol, and in the other it is not. The same bands
have been interpreted by Mirth et al. [2] as originating from a symmetric and
an asymmetric bending of the two hydrogen atoms of the protonated hydroxyl
group of methanol. Obviously, this explanation cannot be used for the similar
bands of a Br�nsted acidic zeolite with acetonitrile around 2400 cm�1 and
2800 cm�1. In liquids broadening of the XH stretch bands upon hydrogen
bonding has been explained from the coupling with XH� � �B intermolecular
stretch modes.[3{5] Splitting of the XH stretch bands has been explained
by Fermi resonances of overtones of XH bending modes.[3, 5] Based on a
comparison of the in-plane bending modes of an acidic OH group in a zeolite
and peak minima in the infrared spectrum, Pelmenschikov et al. [6] proposed
a Fermi resonance of the downwards shifted OH stretch mode and the upwards
shifted in-plane bending overtone.
In order to examine these models in more detail, in the current paper
we present calculations of vibrational frequencies and infrared absorption in-
tensities including anharmonicities. Vibrational wave functions are computed
for the stretch and bending modes of an acidic proton in a zeolite, and of
such a proton with an adsorbed molecule of acetonitrile, where the coupling
of the proton modes with the stretch mode of the acetonitrile molecule as a
whole with respect to the OH group was included. Anharmonic terms up to
fourth order in the potential energy were included, and infrared absorption
intensities were computed using a fourth-order dipole surface. The vibrational
wave functions were computed in a variational approach. The potential en-
ergy and dipole surfaces were �tted to quantum chemical calculations using
polynomials.
Theory 45
3.2 Theory
3.2.1 The Potential Energy and Dipole Surfaces
The computation of the potential energy and dipole surfaces has been done
at the Hartree-Fock level of theory (SCF), and also using density functional
theory (DFT). The density functional applied was Becke's 3 parameter func-
tional with the non-local correlation provided by the Lee, Yang, and Parr
expression.[7{10] DFT is assumed to yield results that are closer to experi-
ment, for the energy as well as for the dipole surface.[11] Since the computed
infrared spectra from the SCF data and the DFT data were qualitatively very
similar, the DFT results are presented completely, and only in some cases
supplemented by the SCF results.
The cluster approach has been used to model the acidic site of the zeolite.
We have used a relatively small cluster, to allow for electronic structure cal-
culations with a reasonably good basis set. The cluster contained the acidic
Al(OH)Si group with the dangling bonds of Al and Si saturated by OH groups.
A restricted geometry optimisation has been done for this cluster, with and
without an interacting molecule of acetonitrile, to obtain reference points for
the potential energy and dipole surfaces. Full optimisation of the geometry of
the cluster was not done for two reasons.
Firstly, the cluster molecule forms internal hydrogen bridges between
the terminating OH groups. These hydrogen bridges a�ect the (Al{O{Si)
angle, and hence the OH-frequency, in a way that is not found in zeolite
systems, which we attempt to model. To prevent internal hydrogen bridging,
we have optimised the zeolite cluster without acetonitrile with the (Si{O{H)
and (Al{O{H) angles of the terminatingOH groups �xed at tetrahedral angles,
and required that, going from the central O atom to a terminal OH, the atoms
O{T{O{H (T=Si or T=Al), should be in one plane.
The second reason not to optimise fully has been computational cost. We
have imposed Cs symmetry, with the Al(OH)Si part of the cluster positioned
in the mirror plane. This has allowed for a considerable reduction of the
number of points to be computed for the potential energy surface, as both
sides of the mirror plane are equivalent.
In the optimisation of the geometry of the zeolite cluster interacting with
acetonitrile, we have �xed the terminal OH groups to the positions found in
the optimisation of the free zeolite cluster, attempting to model the `rigidity'
of the zeolite lattice. The acetonitrile has been positioned with the CN group
pointing towards the acid OH group, as shown in Fig. 1.
46 Chapter 3: Three and Four-Coordinate Cluster Models
O
O
O
O
OO
H
HH
H
HH
AlO
Si
H
N
C
C
HH H
Figure 1. Zeolite cluster with acetonitrile, optimised using DFT.
This mode of adsorption is energetically the most favourable. The two C
atoms and the N atom of acetonitrile have been kept �xed on one line. As
a result of the optimisation, no proton transfer of the zeolite cluster to the
acetonitrile has been observed, nor has a local minimum in the energy for
a proton position closer to the acetonitrile than to the zeolite cluster been
found. A number of geometrical parameters obtained in the optimisations is
given in Tab. 1.
Table 1. Main results of geometry optimisations. Some geometrical parameters of
the central Al{OH{Si group of the zeolite cluster and the N atom of the acetoni-
trile molecule are given. Distances r(: : :) are in �Angstrom, angles in degrees. The
columns denoted `SCF' give the Hartree-Fock results, those denoted `DFT' the den-
sity functional theory results.
zeolite cluster zeolite cluster change
without acetonitrile with acetonitrile after absorption
Both three and the four-dimensional potential energy surfaces, we found
that the coeÆcient of the fourth order term in the OH stretch coordinate
was negative. This implies that there will be non-physical eigenstates of the
Hamiltonian, that will be found if the basis functions used extend into the area
of the coordinate space where the polynomial describing the potential energy
goes to minus in�nity. In the basis set we used, this was not a problem, again
due to the fact that the wave functions of interest were mostly restricted to an
area near the potential energy minimum, thus no basis functions were needed
that extended beyond the sampled coordinate space.
50 Chapter 3: Three and Four-Coordinate Cluster Models
3.2.3 Dynamics
For the vibrational calculations we have described the molecular systems by
a set of linear coordinates that facilitates the interpretation of the vibrational
states in terms of stretch and bending modes. The OH stretch coordinate
describes the movement of the acidic H along the equilibrium OH-bond. The
in-plane bending coordinate describes the movement of the acidic H perpen-
dicular to the OH stretch coordinate, in the (Al{O{Si)-plane of the cluster.
The out-of-plane bending coordinate describes the movement perpendicular
to the OH stretch coordinate, and perpendicular to the (Al{O{Si)-plane of
the cluster as well. Finally, in the calculations with acetonitrile, the acetoni-
trile stretch coordinate describes the movement of the acetonitrile molecule
as a whole with respect to the zeolite cluster, along its (C{C�N)-axis.In the computation of the kinetic energy, all atoms of the zeolite cluster,
except for the acidic H, were kept �xed. The Hamiltonian is given by:[23]
H =1
2
DXi=1
DXj=1
(M�1)ijpipj +X
�1;���;�D
a�1;���;�D
DYi=1
q�ii ; with 0 �
DXi=1
�i � n;
(2)
where D is the number of internal coordinates qi with conjugated momenta
pi, M�1 is the inverse mass matrix, and the a�1;���;�D are the coeÆcients of
the polynomial describing the potential energy. The order n of the potential
energy polynomial was four in the computations in this paper. The way in
which the polynomial is truncated ensures that the shape of the potential
energy surface does not depend on a particular choice of internal coordinates.
Advantages of a polynomial as a functional form are that it has no bias towards
the shape of the potential energy, it generates a sparse Hamiltonian matrix in
the basis set we employed, and allows for eÆcient analytical computation of
matrix elements. A disadvantage is the unphysical behaviour beyond the area
in the internal coordinate space where the �t was made. This can lead to low
energies for basis functions of high order due to their having a non-negligible
amplitude in an area where the value of the polynomial goes to minus in�nity.
Such basis functions should be avoided. The cross terms M�1ij pipj in the
kinetic energy can have non-zero values if the internal coordinates used are
not orthogonal (which is not the case in the current paper).
To �nd the eigenvalues of the vibrational Hamiltonian, we applied the
linear variational principle in which the wave function is expanded in products
of one-dimensional harmonic eigenfunctions (Hermite functions).[24]
(q1; � � � ; qD) =X
�1;���;�D
c�1;���;�D
DYi=1
�(�i)(qi) (3)
Theory 51
In this expression is the vibrational wave function in D dimensions, and
�(�i)(qi) is the normalised �
thi order Hermite function of coordinate qi. This
is similar to the method used by Mijoule et al.[25] One-dimensional harmonic
eigenfunctions are characterised by the ratio of a mass and a force constant.
The masses of the one-dimensional components of the basis functions were
computed taking the inverse of the diagonal elements of the inverse mass ma-
trix (M�1ii in Eq. 2) in our internal coordinate description. The force constants
associated with the Hermite functions can be derived from the curvature at
the coordinate origin of the potential energy surface in the direction of the
corresponding internal coordinate, provided the coordinate origin represents
a minimum in this direction. In the present work this was mostly the case,
except for the out-of-plane bending coordinate of the acidic proton without
acetonitrile, which will be discussed in the Results section.
The vibrational basis set used for the acidic proton with acetonitrile con-
sisted of all products of 4 Hermite functions of which the sum of the orders
did not exceed 12, yielding a total number of 1820 basis functions. We tested
several basis set sizes and concluded that there is very little di�erence be-
tween a spectrum computed with a 10� 10� 10� 10 basis and one computed
with a 12 � 12 � 12 � 12 basis, indicating that the basis set is almost con-
verged. However, there will be no convergence of the computed spectra for
much larger basis sets because the potential has no absolute minimum: it
has a negative fourth order coeÆcient in the OH stretch coordinate. The
12� 12� 12� 12 basis set therefore seems best to describe the wave functionsin the local minimum of our potential, which corresponds to the minimum in
the real potential. For consistency we also applied a 12� 12� 12 basis in the
three-dimensional calculations, with a total of 455 basis functions.
We computed the integrated infrared absorption intensities applying
Fermi's golden rule,[26] and fractional Boltzmann occupation numbers at a
given temperature. The integrated absorption intensities then are given by
Ai!f =2�2�E
3�0h2c2
h��hij�xjfi��2 + ��hij�yjfi��2 + ��hij�zjfi��2i e�Ei=kTPNj e
�Ej=kT: (4)
In this expression the Ai!f is the absorption intensity for one particle per unit
surface, integrated over wave numbers, and averaged over di�erent molecular
orientations, of the transition between initial level i and �nal level f , �0 is the
electrical permittivity of vacuum, c is the speed of light, �E is the di�erence
in energy between the normalised vibrational states jii and jfi, �x, �y, and�z are the components of the dipole operator, Ej is the energy of vibrational
level j, k is the Boltzmann constant, h is Planc's constant, T is the absolute
temperature, and N is the number of levels considered. In the computation
52 Chapter 3: Three and Four-Coordinate Cluster Models
of the transition dipoles hij�jfi we have taken both mechanical anharmonic-
ities (i.e., third order and higher terms in the potential energy) and electric
anharmonicities (i.e., second order and higher in the dipole) into account. Of
these, the mechanical anharmonicities are of far greater in uence.
The energy and dipole operators were converted into a representation of
annihilation and creation operators,[24] to facilitate analytical computation
of matrix elements. The vibrational Hamiltonian eigenvalues and eigenvec-
tors were computed using a Lanczos algorithm, [27, 28] with typically 400
iterations for the three-dimensional computations, and 2400 for the four-
dimensional computations. The described method was implemented in the
AnharmND program, consisting of �6000 lines of C++ code. SVD and Lanc-
zos routines from the Meschach numerical linear algebra library in C were
used.[29]
3.3 Results and Discussion
3.3.1 Acidic Proton without Acetonitrile
In the geometry optimisations of the zeolite cluster, we have imposed through
symmetry restrictions that the acidic proton should stay in the mirror plane
of the cluster. We have computed analytical force constants of the optimised
structure, and found that the force constant of the out-of-plane bending co-
ordinate corresponded to a very low harmonic frequency of 251 cm�1. The
fourth order polynomial �t of the potential energy showed a maximum in the
energy for our optimised structure, and a minimum for a proton position at a
distance of 0.254�A (using DFT) from the mirror plane. The low value of the
out-of-plane bending force constant, and the fact that our �t yields a negative
second derivative in this direction, show that the potential energy surface is
very shallow around the minimum, and is poorly described by a harmonic po-
tential. In solving the Schr�odinger equation one has to integrate the potential
energy over the spatial coordinates. Since the �t describes a extended area
of the potential energy, it is better suited for the calculation of spectra than
the analytical force constant in the minimum of the potential, which only de-
scribes the energy in a very small area. Since the Hamiltonian we employed
is symmetrical with respect to the mirror plane of the cluster, the vibrational
states are symmetric or antisymmetric with respect to the mirror plane. To
interpret the calculated states meaningfully in terms of stretch and bending
modes, we have performed the calculations with a set of basis functions cen-
tered around the proton position of minimum energy in the mirror plane of the
zeolite cluster. Because the �t of the energy had a maximum in the point with
respect to the out-of-plane bending coordinate, we could not use its curvature
in that direction to determine the force constant associated with the Hermite
functions. In order to �nd a good value for this force constant we have done
Results and Discussion 53
a calculation in which the Hermite functions have been centered around one
of the two proton positions for which the energy polynomial had a minimum.
From this calculation we have taken the root mean square displacement of
the ground state in the out-of-plane bending coordinate, and computed the
force constant that a Hermite function with the same width would have. The
numerical results of the calculations with the two di�erently centered basis
sets were not noticeably di�erent. In general it is found that the value of
the force constant associated with a basis function can be varied considerably
without seriously a�ecting the results of the lower levels, provided that the
basis set used is not very small.
For the acidic proton we have calculated the 20 vibrational states with
lowest energy, and the absorption spectrum at a temperature of 298:15K.
We have computed integrated absorption intensities, but no peak widths; the
spectra shown in Fig. 2 and 3 have been obtained by convolution of Dirac delta
functions with normalised gaussian curves of width 10 cm�1. This means that
peak positions and peak surfaces are numerically correct, but peak widths
are arbitrary. Tab. 3 shows the frequencies, integrated absorption intensities,
and assignments of the most important transitions in the spectrum. For the
DFT potential energy a comparison is given with values obtained by the dou-
ble harmonic approach, in which a harmonic potential energy surface and a
linear dipole surface are employed. For the double harmonic calculation we
have placed the coordinate origin on one of the proton positions that repre-
sent a minimum in the potential energy, to avoid computing an imaginary
out-of-plane bending frequency.
In the following discussion we will denote vibrational transitions by two
sets of quantum numbers separated by an arrow: 123! 234 denotes a tran-
sition from a vibrational state that is once excited in the OH stretch mode,
twice in the in-plane bending mode, and three times in the out-of-plane bend-
ing mode, to a vibrational state that is one level higher excited in each mode.
54 Chapter 3: Three and Four-Coordinate Cluster Models
Table 3. Computed IR spectrum of the acidic proton without acetonitrile. The
assignment column shows the quantum numbers of the basis function with the largest
coeÆcient for the involved states; the �rst quantum number refers to the OH stretch,
the second to the in-plane bending, and the third to the out-of-plane bending. � is
the absorption frequency in cm�1 , A the infrared integrated absorption intensity in
km/mol. The intensities are computed taking into account a Boltzmann distribution
at 298.15 K. For the DFT potential energy surface the transitions in this table with
non-zero intensity are indicated in Fig. 2.
SCF DFT DFT
Anharmonic Anharmonic Harmonic
� A � A assignment � A
478 110.6 417 81.7 000 ! 001 472 89.2
571 20.8 525 20.0 001 ! 002 472 18.3
667 1.5 621 1.8 002 ! 003 472 2.8
1049 9.9 941 13.0 000 ! 002 945 0.0
1126 208.2 1009 236.4 000 ! 010 952 229.8
1172 17.1 1048 25.9 001 ! 011 952 23.5
1239 5.4 1145 8.0 001 ! 003 945 0.0
2308 3.0 2075 2.0 000 ! 020 1904 0.0
3779 222.6 3375 197.5 000 ! 100 3609 197.4
3975 20.6 3558 25.2 001 ! 101 3609 20.2
In Fig. 2 the computed spectra derived from the DFT surfaces are shown.
The SCF spectra contain the same transitions, but with higher frequencies,
which is due to the fact that SCF overestimates force constants. Comparing
the anharmonic calculation with the harmonic result, the OH stretch frequen-
cies are lower, the in-plane bending frequencies are higher, and the out-of-
plane bending frequency is higher for the SCF calculation, and lower for the
DFT calculation. Due to anharmonicity of the out-of-plane bending bending
the di�erence in energy between the subsequent excited levels increases, as
shown by the transitions at 417 cm�1 (000 ! 001), 525 cm�1 (001 ! 002),
and 621 cm�1 (002 ! 003). This is caused by positive quartic terms in the
potential energy, that render the walls of the potential steeper. Transitions
like 001! 002 and 002! 003 are, of course, only visible because of thermal
population of the excited levels.
Also visible due to thermal population, is the 001 ! 101 transition
at 3558 cm�1. It is higher in frequency than the 000 ! 100 transition at
3375 cm�1, because if the out-of-plane bending mode is excited once, the ef-
fective Hamiltonian for the OH stretch mode gets a larger second derivative,
compared to the ground state.
In the anharmonic spectrum overtones of the out-of-plane bending at
941 cm�1 and of the in-plane bending at 2075 cm�1 are visible, but clearly
much weaker than the fundamentals.
Results and Discussion 55
a)
02468
1012
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
417
525621 941
1009
10481145
3375
3558
b)
02468
1012
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
wave number
472
952 3609
o.o
.p. b
end
i.p. b
end
stre
tch
Figure 2. Computed spectra of zeolite acidic proton, at T=298.15 K. Wave numbers
are in cm�1 , intensity is in 105m2=mol. Potential energy and dipole surfaces were
computed using DFT. Spectrum a) is computed including anharmonicities, spectrum
b) is computed in the double harmonic approach.
Comparing the results with experimental values for zeolite acidic sites,
the DFT OH stretch frequency is low: 3375 cm�1 compared to ca. 3600 cm�1
found in experiments. There may be various reasons for this. Most important
is the fact that we did not allow the oxygen atom to which the acidic proton
is attached to move. If this oxygen were not �xed, the resulting reduced mass
for the OH stretch mode would be lower, and its absorption frequency would
be higher. Also coupling of lower frequency SiO and AlO stretch modes with
the OH stretch mode can cause an increase in the OH stretch frequency. Fur-
thermore, the chosen cluster model and the used DFT method may in uence
the frequencies computed. It was shown earlier that the proton abstraction
energy of zeolite clusters similar to the one we used only becomes cluster inde-
pendent if the cluster represents a substantially larger part of the zeolite [30].
It is not the aim of the present article to resolve this problem. Since further
investigation of the e�ect of the used basis set, and of full geometry optimi-
sation, would only give more information about the zeolite cluster we used,
we did not pursue this in greater detail. In order to check whether the DFT
method applied could be the cause of the low stretch frequency, we have done
a test calculation computing the OH stretch frequency of a silanol group, in a
simple one-dimensional approach. We performed a geometry optimisation of a
Si(OH)4 molecule, employing a 6-31G** basis set, and the same DFT method
as in our previous calculations. Five points of the potential energy surface
have been computed, expanding one of the (equivalent) OH bonds around the
centre of mass of the OH group, by factors of 0.8, 0.9, 1.0, 1.1, and 1.4. This
is the same as the grid used for the OH stretch mode, the di�erence being
that the oxygen position was not kept �xed this time. Fitting the potential
energy with a fourth order polynomial, we have found a stretch frequency of
56 Chapter 3: Three and Four-Coordinate Cluster Models
3641 cm�1. Adding one point of the potential energy where the OH bond
was stretched by a factor of 1.25 yielded a frequency of 3709 cm�1. This is
in good agreement with the experimental value of ca. 3750 cm�1 quoted in
[31], and shows that the density functional method applied is valid for our
purposes. Seeing this result, one might think that adding the `1.25 point'
may improve our calculations of the acidic site also. This turned out not to
be the case. The reason is probably that the potential had already been sam-
pled better, because the bending modes of the proton had been taken into
account. Due to interference of lattice modes, OH bending modes in zeolites
are experimentally not directly discernible. In the present paper however we
have tried to describe the dynamics of a hydrogen bond in order to gain some
physical understanding of the involved phenomena, and we have not tried to
get quantitative agreement with experiment.
Results and Discussion 57
3.3.2 Acidic Proton with Acetonitrile
In the vibrational calculations of the acidic proton with acetonitrile we have
computed 120 levels. It has been necessary to compute many more levels
than in the three-dimensional case, because the added acetonitrile stretch
mode is much lower in frequency than the proton modes, resulting in a large
number of levels lower in energy than the �rst excited OH stretch level. For
these calculations the proton position in the mirror plane of the cluster does
represent a minimum in the potential energy.
a)
05
10152025
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
90777
1214
1670 253831663184
32563270
33603386
b)
05
10152025
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
119 776
11571693
23842507
2596
26252650
27692787
c)
0
20
40
60
80
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
wave number
122 441 1016
2864
acet
on
itri
le
o.o
.p. b
end
i.p. b
end
stre
tch
Figure 3. Computed spectra of zeolite acidic proton interacting with acetonitrile, at
T=298.15 K. Wave numbers are in cm�1 , intensity is in 105m2=mol. Spectrum a)
is computed using an SCF potential energy and dipole surface, spectra b and c) are
based on DFT data. The spectra a) and b) are computed including anharmonicities,
spectrum c) is computed in the double harmonic approach.
The spectra computed for the acidic proton with acetonitrile adsorbed
are shown in Fig. 3. In this �gure also the SCF spectrum is shown, because it
di�ers in a qualitative way from the DFT spectrum. The di�erences between
the anharmonic and the double harmonic spectra are much larger than in
the spectra of the proton alone. An important contribution to this di�erence
stems from the strong anharmonic coupling between the OH stretch and the
acetonitrile stretch, through which many combination bands become visible
in the infrared spectrum, and give the impression of a broadened OH stretch
band. In Tab. 4 the assignments are given for the most important peaks in the
spectra. We did not list integrated absorption intensities, because most of the
58 Chapter 3: Three and Four-Coordinate Cluster Models
visual peaks are a superposition of transitions of which the low and high level
di�er the same number in the excitation level of the acetonitrile stretch; there
are more than a hundred transitions that give a substantial contribution to
the visible bands. Among the transitions that give substantial contributions,
i.e., at least 1 to 10 km/mol, the acetonitrile stretch may be excited as high
as to the seventh level.
Table 4. Computed IR spectrum of the acidic proton with acetonitrile. The as-
signment column shows the quantum numbers of the basis function with the largest
coeÆcient for the involved states; the �rst quantum number refers to the OH stretch,
the second to the in-plane bending, the third to the out-of-plane bending, and the
fourth to the acetonitrile stretch. � is the absorption frequency. All wave numbers
not in brackets are printed in Fig. 3. The values in brackets represent transitions
of little or no absorption intensity.
SCF DFT DFT
Anharmonic Anharmonic Double Harmonic
�(cm�1) �(cm�1) assignment �(cm�1)
90 119 0000 ! 0001 122
(774) 776 0001 ! 0011 441
777 (783) 0000 ! 0010 441
1214 1157 0000 ! 0100 1016
1670 1693 0000 ! 0020 (883)
2538 2384 0000 ! 0200 (2031)
(2708) 2596 0000 ! 0202 (2275)
3166 2507 0001 ! 1000 (2742)
3184 (2395) 0002 ! 1001 (2742)
3256 2625 0000 ! 1000 2864
3270 2650 0001 ! 1001 2864
3360 2769 0000 ! 1001 (2986)
(3362) 2787 0001 ! 1002 (2986)
3386 (2811) 0002 ! 1003 (2986)
Again it is seen that the anharmonic calculations generate a lower OH
stretch frequency compared to the harmonic calculations. All bending fre-
quencies are higher in the anharmonic calculations, and this e�ect is particu-
larly strong for the out-of-plane bending, which almost doubles in frequency
compared to the harmonic calculations.
A very marked change occurring upon acetonitrile adsorption is the in-
crease in infrared absorption intensity. Note the di�erent scales used on the
intensity axes comparing Fig. 2 and 3. This is caused by enhanced polarisa-
tion of the OH bond. For the DFT dipole surface, the component of the dipole
in the direction of the OH bond is �0:48 ea0 for the zeolite cluster without
Results and Discussion 59
acetonitrile, and 2:15 ea0 for the zeolite cluster with acetonitrile. The gradient
of the dipole in the OH bond direction increases from 0:46 e to 1:52 e upon
acetonitrile adsorption. Because the dominant term in the infrared absorp-
tion intensity of a transition between two adjacent OH stretch levels contains
the square of this gradient, an increase of the intensity by roughly a factor
of ten is to be expected. This is seen most clearly comparing the harmonic
spectra in Fig. 2 and 3, because they have all intensity due to the OH stretch
concentrated at a single wave number, whereas in the anharmonic spectra it
is spread over a greater number of peaks.
The changes in the infrared spectrum brought about by acetonitrile ad-
sorption are considerably larger for the DFT calculation than for the SCF
calculation. This con�rms our expectation that the DFT potential energy
surface should yield a spectrum that is closer to experiment. In Tab. 5 the
frequency shifts of the OH stretch and bendings are given. The shift of the
OH stretch appears to be not very sensitive to anharmonicities. The upward
shifts of the bendings are much larger in the anharmonic calculations.
Table 5. Frequency shifts upon acetonitrile adsorption in cm�1 .
stretch in-plane bending out-of-plane bending
SCF anharmonic �523 + 88 +299
SCF harmonic �527 � 36 + 24
DFT anharmonic �760 +148 +359
DFT harmonic �745 + 64 � 31
In the following discussion transitions between vibrational states are de-
noted in the same way as in the previous section, with a fourth quantum
number added to describe excitations in the acetonitrile stretch mode. In the
four-dimensional calculations the overtones of the out-of-plane bending are
visible with enhanced intensity at 1670 cm�1 (SCF) and 1693 cm�1 (DFT).
In the calculation with the SCF potential, the main OH stretch (0000! 1000)
peak at 3256 cm�1 is anked by di�erence (0001 ! 1001) and combination
(0000 ! 1001) bands with the acetonitrile stretch mode, and separate from
the overtone of the in-plane bending at 2538 cm�1. In the four-dimensional
DFT calculation the main OH stretch peak at 2625 cm�1 is anked on the high
frequency side by combination bands with the acetonitrile stretch, whereas on
the low frequency side the overtone of the in-plane bending and this overtone
combined with a doubly excited acetonitrile stretch yield the most impor-
tant peaks (0000 ! 0200 and 0000 ! 0202). The latter peaks have a rela-
tively large intensity due to the fact that the strongly absorbing OH stretch
mode mixes with the in-plane bending overtone vibrational states. This hap-
pens because the frequencies of the OH stretch and the in-plane bending
overtone are moving into each other's direction upon acetonitrile adsorption:
60 Chapter 3: Three and Four-Coordinate Cluster Models
the in-plane bending mode moves to a higher frequency because acetonitrile
pulls the proton in the direction of the mirror plane of the cluster, and the
OH stretch frequency decreases because the interaction with acetonitrile weak-
ens the OH bond. The SCF potential energy surface reproduces this e�ect
less well, so that there is not much mixing of the OH stretch mode with the
in-plane bending overtone, causing the intensity of the 0000! 0200 transition
at 2538 cm�1 (Fig. 3) to have a much smaller value than the corresponding
peak in the DFT spectrum. The interaction of the OH stretch and the over-
tone of the in-plane bending in the DFT calculation can also be seen from
the coeÆcients of the basis functions in the vibrational wave functions that
represent the excited states of the peaks at 2384 cm�1, 2596 cm�1, 2625 cm�1,
and 2650 cm�1. These coeÆcients are listed in Tab. 6. All the states listed
appear to be mixtures of OH stretch and in-plane bending overtone modes.
Table 6. Largest coeÆcients of basis functions of 4 vibrational states. Of four fre-
quencies in Tab. 4 the largest coeÆcients of basis functions of the higher level of the
transition are given. The basis functions are labelled by the order of Hermite func-
tion of OH stretch, in-plane bending, out-of-plane bending, and acetonitrile stretch
coordinate respectively. The coeÆcients of the basis functions are normalised, as
well as the basis functions themselves.
2384 cm�1 2586 cm�1 2625 cm�1 2650 cm�1
function coe�. function coe�. function coe�. function coe�.
where �� are the dipole operator components. The absorption intensity Ai of
the transition from j�0i to j�ii is proportional to the square of the transitiondipole moment and the energy di�erence between the involved states, so that
Ai / ("i � "0)�X�
��h�ij��j�0i��2 = ("i � "0) c2i1X�
p2� (7)
where "i denotes the eigenvalues of j�ii. For the relative intensities we can
write, using c221 = 1� c211 (from the orthogonality of the transformation ma-
trix):
A1PiAi
=("1 � "0) c211
P� p
2�
("1 � "2) c211P
� p2� + ("2 � "0)
P� p
2�
=�1
(�1 � �2) + �2=c211
� f1;
(8)
where �i (/ "i � "0) is the frequency of the transition j�0i ! j�ii. This
enables us to compute c211 from information that can be obtained from either
an experimental, or a computed spectrum:
c211 = f1 �
�2
�1 + (�2 � �1)=f1: (9)
Using
Hj�ii = "ij�ii and j ji =Xk
ckjj�ki; (10)
and the orthogonality of the transformation matrix between the coupled and
the decoupled states, we can compute
h 1jHj 1i = c211("1 � "2) + "2
h 2jHj 2i = c211("2 � "1) + "1;
(11)
and �nally for the frequencies
�j�0i!j 1i =�2(�1 � �2)f1
�1 + (�2 � �1)=f1+ �2
�j�0i!j 2i =�2(�2 � �1)f1
�1 + (�2 � �1)=f1+ �1:
(12)
68 Chapter 3: Three and Four-Coordinate Cluster Models
References
[1] L. Kubelkov�a, J. Nov�akov�a, and K. Nedomov�a; J. Catal., \Reactivity
of Surface Species on Zeolites in Methanol Conversion", 124, 441{450
(1990).
[2] G. Mirth, J. A. Lercher, M. W. Anderson, and J. Klinowski; J. Chem.
Soc. Faraday Trans., \Adsorption Complexes of Methanol on Zeolite
ZSM-5", 86, 3039{3044 (1990).
[3] M. F. Claydon, and N. Sheppard; Chemical Communications, \The Na-
ture of \A,B,C"-type Infrared Spectra of Strongly Hydrogen-bonded Sys-
tems; Pseudo-maxima in Vibrational Spectra", 1431{1433 (1969).
[4] S. E. Odinokov, and A. V. Iogansen; Spectrochim Acta, \Torsional
ets because it is not used in all the embedded potentials we computed. If
we include the MM electrostatic potential in the QM Hamiltonian as de-
scribed earlier, this term is set to zero, because it is part of EQM(polarised).
For further analysis of the results, we have computed potentials where the
MM electrostatic potential was not included in the QM Hamiltonian. Then
Eint
MM(electrostatic) contains the electrostatic interactions of the MM atomic
charges with the Si, Al, O, and acidic H atoms of the QM cluster, and a
correction for the way the charges at the QM/MM junction are dealt with in
EMM(zeolite). The QM Si atoms are assigned a charge of +1:2, the O atoms
a charge of �0:6, both like the MM atoms. The QM Al atom gets a charge of
+1:0, and the QM H atom of the acid OH group gets a charge of +0:2. The
procedure where the charges on the oxygen atoms at the QM/MM junction
were shifted to their nearest neighbours, which also a�ects the EMM(zeolite)
term in Eq. 2, has not been used for the potentials without the electrostatic
potential in the QM Hamiltonian. Additionally, if Eint
MM(electrostatic) is used,
the charge on the junction MM oxygen atoms is �0:6, instead of �0:6=2 whenthere is only one silicon neighbour with a MM point charge. For this discussion
we will formally assume that Eint
MM(electrostatic) contains terms to represent
these di�erences in EMM(zeolite). The atoms of acetonitrile do not contribute
to Eint
MM(electrostatic).
Four di�erent potentials were computed from the same geometries as the
fully embedded potential. Their contributing terms are listed in Tab. 2.
Computational Details 103
Table 2. Contributing terms to the total energy in the di�erent embedding levels.
The �rst row denotes the di�erent levels of embedding used in this study, the �rst
column lists the di�erent contributions. A `+' indicates a certain contribution is
added, a `�' indicates it is subtracted. The contributions are described in the text.
none mechanical electrostatic full
EQM(polarised) + +
EQM(non-polarised) + +
EMM(zeolite) + +
Eint
MM(junction) + +
Eint
MM(repulsion) + +
Eint
MM(acetonitrile) + +
Eint
MM(electrostatic) + �
The geometries for the calculation of the potential energy surfaces were de-
rived from the optimized geometry obtained in what we call `full' embedding.
This corresponds to Eq. 2 with EQM = EQM(polarised) and is summarized in
the fourth column of Tab. 2. Another potential energy surface is derived with-
out any embedding interactions, this is shown in the �rst column of Tab. 2.
Then there is a level of embedding where all the interactions between the
QM and the MM part are computed by molecular mechanics, which we call
`mechanical embedding'. As shown in the second column of Tab. 2, this in-
cludes the Eint
MM(electrostatic) term described above. Finally, we isolate the
e�ect of the MM electrostatic potential added to the QM Hamiltonian by
adding the di�erence of the full embedded potential with the mechanically
embedded potential to the potential without any embedding. The result is
summarized in the third column of Tab. 2. Note that to obtain four di�erent
potential energy surfaces, two QM-type surfaces had to be computed, one
with EQM(non-polarised), and one with EQM(polarised).
5.2.3 The Calculation of the Infrared Spectra
The coordinates used for the polynomials �tting potential energy and in the
vibrational calculations were the following. For the acidic hydrogen atom
the stretch coordinate was de�ned as the linear movement in the direction
of the equilibrium OH bond, the in-plane bending as the linear displacement
of the hydrogen atom perpendicular to the stretch, in the plane of the Si{
O{Al atoms of the Br�nsted site, and the out-of-plane bending as the linear
displacement of the hydrogen atom perpendicular to both the stretch and
the in-plane bending coordinate. For the adsorbed complex we additionally
de�ned an acetonitrile stretch coordinate as the linear displacement of the
acetonitrile molecule along its C{C�N axis.
104 Chapter 5: Cluster versus Embedded Model
The Hamiltonian employed in the vibrational calculations has the follow-
ing form:
H =1
2
DXi=1
DXj=1
(M�1)ijpipj +X
�1;:::;�D
a�1;:::;�D
DYi=1
q�ii ; with 0 �
DXi=1
�i � N
(6)
In this expression the D is the number of dimensions, qi are the spatial co-
ordinates with conjugated momenta pi, M�1 is the inverse mass matrix, and
a�1;:::;�D are the coeÆcients of the polynomial representing the potential en-
ergy. The polynomials used to �t the potential energy in this paper have all
been of fourth order, which corresponds with N = 4 in Eq. 6.
The wave functions were expanded in products of one-dimensional har-
monic eigenfunctions (Hermite functions).[23]
(q1 : : : qD) =X
�1;:::;�D
c�1;:::;�D
DYi=1
�(�i)(qi) (7)
Here is the D-dimensional vibrational wave function, and �(�i)(qi) is the
normalized �thi order Hermite function of coordinate qi. The basis has been
truncated by specifying a maximum order �(max)
i for each coordinate, and
subsequently limiting the basis functions used to those that meet the following
condition:DXi=1
�i
�
(max)
i
� 1 (8)
In the computations described here �(max)
i = 12 for all the coordinates. For
the three- and four-dimensional dimensional calculations this yields basis sets
of 455 and 1820 functions respectively.
Infrared absorption intensities were computed using Fermi's golden rule
[24] and fractional Boltzmann occupation numbers at a given temperature.
The integrated absorption intensity Ai!f from initial level i to �nal level f
is given by
Ai!f =2�2�E
3�0h2c2
X�=x;y;z
��hij��jfi��2 e�Ei=kTPLj e
�Ej=kT(9)
In this expression Ai!f is de�ned for one particle per unit surface, integrated
over wave numbers and averaged over di�erent molecular orientations. �E is
the di�erence in energy between the normalized states jii and jfi, �0 is the
electrical permittivity of vacuum, h is Planck's constant, c is the speed of
Results and Discussion 105
light in vacuum, �x, �y, and �z are the components of the dipole operator,
Ej is the energy of vibrational level j, k is the Boltzmann constant, T is the
absolute temperature, and L is the number of levels taken into account.
In the calculation of matrix elements of the Hamiltonian and of the
dipole component operators, mechanical as well as electric anharmonicities
were taken into account. The vibrational calculations were carried out with
the AnharmND program written by E.L.M. For a more detailed discussion of
the method we refer to our previous papers on the subject. [4, 5]
5.3 Results and Discussion
The �gures of infrared spectra in this paper have been obtained by convo-
luting delta functions representing the calculated intensity and position for
the transitions, with Gaussian curves of half width 10 cm�1 and unity surface
area. As a consequence the surface area under the graphs correctly represents
the calculated integrated absorption intensity, whereas the peak widths are
arbitrary. The alternative of plotting spikes where the height represents the
absorption intensity is not suitable to display spectra with a high density of
peaks, like the infrared spectra with acetonitrile. All spectra have been com-
puted taking into account a Boltzmann distribution over the calculated states
at a temperature of 298.15K.
In the plotted infrared spectra we have printed the wave numbers of im-
portant transitions. For the spectra of the zeolite OH without acetonitrile
these numbers have been derived from di�erences between computed levels.
For the spectra of the zeolite OH with acetonitrile, there are many overlap-
ping hot bands, because higher excited states of the acetonitrile mode are
populated signi�cantly at room temperature. Therefore we have printed wave
numbers that correspond to the maxima in the plotted graphs. They represent
a weighted average for di�erent transitions that contribute to the peaks.
5.3.1 Basis Set E�ects
In previous work we have computed the infrared spectra of a zeolite OH with
or without acetonitrile adsorbed in a cluster approach.[4, 5] In this paper we
are comparing the results obtained from `bare' clusters with those we get from
embedded clusters. The mixed basis set we used before has high quality basis
functions on the central OH group and the nitrogen atom of acetonitrile (6-
311+G**), low quality functions on the terminal hydrogen atoms (STO-3G),
and 6-31G** on the other atoms of the cluster. For the embedded calculations
we have used 6-31G* throughout, which resulted in a smaller overall basis set.
Fig. 2 and Fig. 3 show the spectra as computed with bare clusters, with the
mixed and the balanced 6-31G* basis set respectively.
106 Chapter 5: Cluster versus Embedded Model
a)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
4781126 3779
b)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
wave number
539 11471163 3741
Figure 2. The infrared spectrum of the zeolite OH group without acetonitrile, cal-
culated in a cluster approach with a mixed basis set (a) and a balanced basis set
(b). Frequencies on the horizontal axis are in cm�1 , absorption intensities on the
vertical axis are in 105m2=mol.
For the zeolite clusters without acetonitrile the printed wave numbers
from left to right are for the out-of-plane bending, in-plane bending, and
stretch mode respectively. The mixed basis set spectrum has lower frequency
bending modes and a higher frequency stretch. The higher stretch frequency
can be understood by a better description of the OH bond, whereas the lower
bending frequencies re ect the greater exibility from the basis set used on
oxygen.
For the in-plane bending of the spectrum with the balanced basis set
(Fig. 2b) there are two wave numbers printed. In this particular spectrum
a Fermi resonance occurs between the in-plane bending and the out-of-plane
bending overtone. As a result there are two peaks of approximately equal
intensity at 1147 and 1163 cm�1, which overlap in the �gure. In Fig. 2a the
out-of-plane bending overtone does not interact with the in-plane bending,
and is visible as a very small peak just to the left of the in-plane bending at
1126 cm�1.
Results and Discussion 107
a)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
87776
1213
1672 25363220
3258
3363
b)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
wave number
90772
1221
1653 25533274
3357
3463
Figure 3. The infrared spectrum of the zeolite OH group with acetonitrile, calculated
in a cluster approach with a mixed basis set (a) and a balanced basis set (b). Fre-
quencies on the horizontal axis are in cm�1 , absorption intensities on the vertical
axis are in 105m2=mol.
For the zeolite cluster with adsorbed acetonitrile the wave numbers from
left to right refer to the acetonitrile stretch mode, the OH out-of-plane bending
and in-plane bending modes, followed by their respective �rst overtones, and
the OH stretch mode along with di�erence and combination modes with the
acetonitrile stretch. The upward shift of the bending modes is larger for the
mixed basis set, and the absolute frequency of the OH stretch is lower than for
the balanced basis set, although it was higher before acetonitrile adsorption.
Both the upward shift of the in-plane bending and the downward shift of
the OH stretch are indicative of the strength of the interaction between the
OH group and acetonitrile. The argument for using the enlarged basis set
on the acid OH group and N has been that it yields a better description of
the interaction between the zeolite OH group and acetonitrile.[4, 25, 25] From
Fig. 2 and Fig. 3 we conclude that it does exactly that.
5.3.2 Embedding E�ects
In Fig. 4 the spectra of the zeolite OH without acetonitrile from the embed-
ded cluster have been plotted, with di�erent levels of interaction between the
quantum mechanical cluster part (QM) and the molecular mechanics embed-
ding (MM). Fig. 4a represents the spectrum of the embedded cluster without
any interaction between QM and MM parts, Fig. 4b and 4c are the spectra
with only the mechanical embedding and with only the electrostatic interac-
tion respectively, and Fig. 4d is the spectrum of the embedded cluster with
all interactions between QM and MM parts.
108 Chapter 5: Cluster versus Embedded Model
a)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
4551066
3737
b)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
4551067
3737
c)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
4661126
3724
d)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
wave number
4661126
3724
Figure 4. The infrared spectrum of the zeolite OH group without acetonitrile, calcu-
lated in an embedded cluster approach with di�erent levels of interaction between
the quantum mechanical part and the molecular mechanics part. Spectrum (a) is
calculated without interaction between the QM and the MM part, (b) only has the
mechanical embedding interactions, (c) only has the electrostatic interaction of the
MM part, and (d) has all the interactions between the QM part and the MM part.
Frequencies on the horizontal axis are in cm�1 , absorption intensities on the vertical
axis are in 105m2=mol.
A comparison between Fig. 2b and Fig. 4a shows that the bending modes
have lower frequencies for the cluster used in the embedded calculations than
for the bare cluster used in the previous work.[4] For the in-plane bend the fact
that in the bare cluster the acidic OH group has an eclipsed con�guration with
respect to the terminating OH groups on Al and Si, whereas for the embedded
cluster it is staggered with respect to the SiH groups, could be o�ered as an
explanation for the di�erence in the in-plane bending frequency. There are
also di�erences in chemical composition between the bare and the embedded
cluster, and in the embedding the SiH bonds were �xed at a distance of
1.5�A. In a cluster in which the Si{H distances are optimized, they would be
somewhat smaller than the �xed distance of 1.5�A used here. It has been shown
Results and Discussion 109
that the acid strength of Si{H terminated aluminosilicate clusters is sensitive
to the Si{H distance.[12, 26, 26] This accounts for the lower frequencies in the
embedded cluster (Fig. 4a) compared to the bare cluster (Fig. 2b).
The di�erent parts of Fig. 4 reveal that almost all of the di�erence be-
tween the fully embedded spectrum (Fig. 4d) and the spectrum of the embed-
ded cluster without any interaction between QM and MM part (Fig. 4a) is
due to the electrostatic interaction of the QM part with the MM part. The ef-
fect of the mechanical embedding is negligible, because only the non-bonding
interaction terms in it can a�ect the computed infrared spectrum. These non-
bonding interactions between the acidic proton and the surrounding zeolite
are very small indeed.
To explain why the electrostatic interaction a�ects the in-plane bend sig-
ni�cantly more than the out-of-plane bend, we inspected two cross-sections of
the applied electrostatic potential. Fig. 5a shows contours of the electrostatic
potential in the region of the acid site for the optimized embedded cluster
with acetonitrile. The contours are shown at intervals of 0.0025 a.u., and a
cross-section is cut in the plane approximately containing the O{H and N{C
dipoles. For the free acid, embedded in the full potential, the electrostatic
potential over the site may be slightly di�erent, but not in a qualitative way.
Observing the curvature of the contours around the O{H dipole it can be
seen that the in-plane bend will be hindered by the electrostatic �eld: the
contours curve against the O{H dipole, thereby introducing an unfavourable
interaction when moving the O{H dipole in the O{Al{O plane.
Fig. 5b shows that the electrostatic potential almost curves along the
out-of-plane bend of the acid O{H group. As a result the added potential for
this mode is shallow and the e�ect on the frequency of the out-of-plane bend
is small.
110 Chapter 5: Cluster versus Embedded Model
Figure 5. Cross-sections of the electrostatic potential applied to the optimized ace-
tonitrile on an acid site; cross-section (a) is in the plane approximately parallel to
the O{H and N{C dipoles, and (b) is in a plane perpendicular to the one in (a).
Contours are at intervals of 0.0025 a.u.
In Fig. 6 the infrared spectra computed in the harmonic approximation
of the embedded cluster without acetonitrile are shown.
a)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
4341011 3873
b)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
wave number
4641060
3882
Figure 6. The infrared spectrum of the zeolite OH group without acetonitrile, calcu-
lated in an embedded cluster approach with (a) no, and (b) full interaction between
the QM and MM parts. The infrared spectra were calculated in a harmonic approx-
imation. Frequencies on the horizontal axis are in cm�1 , absorption intensities on
the vertical axis are in 105m2=mol.
Results and Discussion 111
Comparing the harmonic spectrum in Fig. 6a with its anharmonic coun-
terpart in Fig. 4a, the OH stretch frequency is higher and the bending mode
frequencies are lower, but qualitatively the spectra are very much alike. Note
that anharmonicity does not always lower frequencies. The e�ect of the em-
bedding interactions between the QM and the MM part is larger for the
harmonic out-of-plane bend than for the anharmonic out-of-plane bend, but
smaller for the in-plane bend, which is more important for the Fermi-resonance
we seek to reproduce. The shift of the harmonic stretch due to the embedding
interactions has the opposite sign of that found in the anharmonic spectrum.
For the zeolite OH group with adsorbed acetonitrile the spectra of the
embedded clusters at di�erent levels of interaction are given in Fig. 7. Here
the di�erences are more pronounced than in the case without acetonitrile.
Firstly there is a clear e�ect of the di�erent cluster optimized in the em-
bedded approach (Fig. 7a) compared to the bare cluster approach (Fig. 3b).
The embedded cluster has a di�erent chemical composition, and it has been
optimized to mimic a site in an actual zeolite. This results in a cluster that
has a stronger interaction with acetonitrile than then bare cluster, which
is apparent in a number of peaks in the spectra. The bending modes are
higher in frequency for the embedded cluster. As a consequence the in-
plane bend overtone also is higher for the embedded cluster (2936 cm�1 vs.
2553 cm�1), and it comes closer to the OH stretch which is lower in the em-
bedded cluster.(3334 cm�1 vs. 3357 cm�1) This is the e�ect we are after; if the
in-plane bend overtone and the OH stretch coincide, Fermi resonance between
the two modes occurs. The acetonitrile stretch frequency is slightly higher in
Fig. 7a than in Fig. 3b (94 cm�1 vs. 90 cm�1), which also hints at a stronger
interaction.
In the comparison between the spectra computed from the bare cluster
with the embedded cluster without additional interactions (Fig. 2b vs. Fig. 4a,
and Fig. 3b vs. Fig. 7a), we �nd lower OH bend frequencies in the absence, and
higher OH bend frequencies in the presence of acetonitrile. Both di�erences
can be accounted for assuming that the embedded acetonitrile cluster is more
acidic than the bare cluster with the balanced basis set. The embedded cluster
can be more acidic because the residual negative charge can be spread over
a larger volume as the Al atom is surrounded by four O{SiH3 groups. As
pointed out in the paragraph on the clusters without acetonitrile, the �xed
SiH distances also play a role in the strength of the acid OH bond.
We have two indications that the embedded cluster with mechanical
and electrostatic interactions is more acidic than the (HO)3Si{(OH){Al(OH)3cluster. Firstly the adsorption energy of acetonitrile on the bare cluster is
�22:4 kJ=mol, and that on the embedded cluster is �76 kJ=mol. These num-bers include deformation energies of acetonitrile and the zeolite cluster; they
are not corrected for basis set superposition errors (BSSE). Secondly it has
112 Chapter 5: Cluster versus Embedded Model
been shown that there is a correlation between the barrier for H/D exchange
of methane and the proton aÆnity of a cluster.[26] A higher barrier for the
D/H exchange reaction corresponds to a lower proton aÆnity, which in turn
corresponds to a higher acidity. At Hartree-Fock level, the `trimer' cluster
used in Ref. 26 had a barrier of 230 kJ=mol with a 6-31G** basis set, and
the chemically identical cluster in Ref. 12 had a barrier of 250 kJ=mol with a
6-31G* basis set in the embedding scheme we also use in the present paper.
This indicates that the embedding renders the cluster more acidic.
a)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
94788
1460
1683 2936 3249
3334
3443
b)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
80 732
1426
28513302
3368
3460
c)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
101776
1620
1756
3276
3494
d)
0
10
20
0 500 1000 1500 2000 2500 3000 3500 4000
inte
nsi
ty
wave number
93773
1591
16453266
3436
Figure 7. The infrared spectrum of the zeolite OH group with acetonitrile, calcu-
lated in an embedded cluster approach with di�erent levels of interaction between
the quantum mechanical part and the molecular mechanics part. Spectrum (a) is
calculated without interaction between the QM and the MM part, (b) only has the
mechanical embedding interactions, (c) only has the electrostatic interaction of the
MM part, and (d) has all the interactions between the QM part and the MM part.
Frequencies on the horizontal axis are in cm�1 , absorption intensities on the vertical
axis are in 105m2=mol.
Haw et al.[27] have modelled the Br�nsted acid site in ZSM-5 replacing
the silicon atom on the T6 position with a cluster containing one aluminium
Results and Discussion 113
atom and two neighbouring silicon atoms. They have computed an interaction
energy of �56:1 kJ=mol at Hartree-Fock level without correction for BSSE.
The NH distance they �nd is 1.56�A, which is rather short compared to the
1.93�A we �nd for our bare cluster with balanced basis set, and 2.04�A for our
embedded cluster. It may well be that their cluster is somewhat too reactive
because they �xed all silicon, aluminium, and oxygen atoms at positions taken
from crystallographical data. These positions most likely do not correspond to
a minimum in energy for the method they used. Barbosa et al.[28] computed
interaction energies for acetonitrile with a zeolite cluster of �42:6 kJ=mol atHartree-Fock level, and �47:5 kJ=mol at DFT level without BSSE correction,
and of �38 kJ=mol at DFT level with full counterpoise correction.[29] The NH
distances they found were 1.88�A for Hartree-Fock and 1.73�A for DFT. Their
cluster contains two hydrogen terminated silicon atoms and one hydroxyl ter-
minated aluminium atom. The somewhat smaller NH distance and somewhat
larger adsorption energy compared to our bare cluster can be explained by
the di�erence in cluster, and the larger basis set they used.
Experimental heats of absorption have been measured for acetonitrile on
Br�nsted acidic groups in the range from �60 to �80 kJ=mol (depending onloading) in zeolite HY[30] and of �110 kJ=mol in H-ZSM-5.[31] Part of the
di�erence between these experimental values can be ascribed to the smaller
pore size of ZSM-5; measurements of adsorption heat of acetonitrile in silicious
ZSM-5 yield a value that is 10 kJ=mol larger than in silicious faujasite.[30] The
rest is probably due to the di�erence in Si/Al ratios. The ZSM-5 sample of
Ref.[31] has a Si/Al ratio of approximately 25, whereas the Si/Al ratio of the
HY sample of Ref.[30] is only 2.4. The acidic strength of Br�nsted sites in
zeolites decreases with higher aluminium content.
After correction for BSSE, the interaction energy of the bare cluster with
acetonitrile becomes �17:0 kJ=mol. This means the BSSE constitutes 24% of
the interaction energy. Since this is quite substantial we have done a few tests
to assess the e�ect of BSSE on computed frequencies. First we have computed
the interaction energy and the BSSE of the bare zeolite cluster with acetoni-
trile at equilibrium distance, and 0.15�A closer and further away. For these
three points we �tted the interaction energy and the counterpoise corrected
interaction energy with a second order polynomial. We derived that the coun-
terpoise correction decreases the harmonic vibration frequency by 3.3%, and
elongates the distance between the zeolite cluster and acetonitrile by 0.05�A.
The counterpoise corrected interaction energy at the displaced minimum is
�17:3 kJ=mol. Because the main interest of this article is in the vibrations
of the acidic OH group, we also checked the e�ect of a counterpoise correc-
tion one a one-dimensional stretch vibration of the acidic hydrogen atom in
the presence of acetonitrile at equilibrium distance. A fourth order potential
for the OH stretch was computed from points at 0.8, 0.9, 1.0, 1.1 and 1.4
114 Chapter 5: Cluster versus Embedded Model
times the equilibrium OH bond length. The counterpoise correction causes
a decrease in the computed anharmonic fundamental frequency of 0.6%, and
an increase of the expectation value of the OH distance of 0.04%. We con-
clude that counterpoise correction does not greatly in uence the computed
spectrum. Note that to apply a correction for BSSE rigorously, the cost of
computing a potential energy surface easily increases by a factor of three to
four. It is also not clear how to deal with dipole surfaces in a way that is
consistent with a counterpoise correction.
The two di�erent types of interaction between the QM and the MM part
have opposite e�ects on the spectrum of the zeolite OH with adsorbed acetoni-
trile. The e�ect of the mechanical embedding on this spectrum is noticeable,
because acetonitrile has signi�cant non-bonding interactions with the sur-
rounding zeolite. These interactions pull the acetonitrile molecule closer to
the zeolite wall, and away from the acidic proton. In Tab. 3 some expectation
values of atomic distances and angles are listed, and it can be seen that for
the embedded cluster with mechanical interactions, the ON and NH distances
are larger than for the embedded cluster without any interactions. The OH
distance of the acidic group on the other hand is shorter for the cluster with
mechanical interactions. These geometrical di�erences indicate that the em-
bedded cluster with mechanical interactions has a smaller interaction with ace-
tonitrile, and this is con�rmed by the di�erences between the spectra in Fig. 7a
and Fig. 7b: the OH stretch frequency goes up (3334! 3368 cm�1), and the
bending frequencies go down (788! 732 cm�1 and 1460! 1426 cm�1), i.e.,
all OH frequencies shift in the direction of the values they have in the spec-
trum without acetonitrile. Also the acetonitrile stretch frequency is smaller
in Fig. 7a than in Fig. 7b, in accordance with a smaller interaction.
Table 3. Expectation values for distances and angles between the oxygen and hy-
drogen atoms of the acid OH group, and nitrogen of the acetonitrile molecule, for
di�erent levels of embedding interactions included. The angle 6 (Al{O{N) has the
same value of 120.55Æ for all interaction levels. Distances are in �A, and angles in
degrees.
interactions rOH rON rNH 6 (Al{O{H) 6 (H{O{N) 6 (Al{O{Si{H)
In de�nition (2) the height of the potential energy cup is speci�ed at the bor-
ders of the sampled area of the potential energy surface. The PotCup keyword
assumes atomic units always. It cannot be used with input = polynom.
If a potential cup is used, the program �rst �ts the potential in the same
way it would do otherwise, and then shifts the origin to the minimum. Then it
subtracts the value of the potential energy cup from the input energy points,
and �ts the potential energy again. The polynomial �tted to this corrected
surface is added to the potential energy cup function to yield the potential
energy that will be used in the Hamiltonian. From this it will be clear that
it makes no sense to specify a potential cup of an order not higher than the
maximum order of the polynomial that �ts the potential energy. Note that it
is possible to give one or more coeÆcients of the potential cup a value of 0.
The program will print average values of the correction applied to the
input energies, in order to provide an idea of how big its in uence will be.
Also it computes the expectation values of the `cupless' Hamiltonian of the
wave functions, so they can be compared with the eigenvalues. In the ideal
case there should be very little di�erence. The higher the order of the potential
energy cup, the atter it will be near the origin, and steeper further away. It
seems therefore best to make this order very high. However, you pay for this in
computation time, since the number of non-zero elements in the Hamiltonian
will grow.
Adjusting computational parameters 139
Warning: I have not used the PotCup feature a lot. It is possible that it
does not operate well with some of the input possibilities. Be sure to check
if the output is reasonable.
A.9 Adjusting computational parameters
A.9.1 Control of the �t procedure
The keyword FitSelect controls the way the �ts of energy and dipole surface
are done. It has three possible options. The default value is order, which
implies the program will use coeÆcients up to the order speci�ed by the user
in the internal coordinate speci�cation. If orders of the internal coordinates
are not all equal, the program chooses the coeÆcients in such a way that
a `weighted total order' does not exceed the speci�ed orders. If the option
FitSelect = automatic is speci�ed, AnharmND will try to �nd a number
of coeÆcients as close as possible to the number of input points. The option
FitSelect = user allows the user to specify explicitly which coeÆcients to
use. First the number of coeÆcients should be given, then the powers of the
coordinates of the terms that should be included in the �t. For H2O this could
be:
FitSelect = user
4 # going to use only 4 coefficients
0 0 0
2 0 0
0 2 0
0 0 2
With internal coordinates fqig this would specify a potential energy of the
form
c000 + c200 q21 + c020 q
22 + c002 q
23 :
If the constant term is not speci�ed by the user, it will be included by the
program.
The �ts of energy and dipole surfaces are done by a least squares �t.
For each point in the input a linear equation in the polynomial coeÆcients
is set up. This yields a set of equations Ax = b, where x are the values of
the energy or dipole component, the rows of A the corresponding products
of internal coordinates, and x the polynomial coeÆcients. A singular value
decomposition (SVD) of A is then performed, yielding A = UT�V. � is a
diagonal matrix which is uniquely de�ned under the condition that its values
are non-negative and non-increasing down the diagonal. If this is so, its values
f�ig are called singular values. For least squares problems like this one, the
condition number of the matrix is given by �1=�n, if A is m� n and m � n.
AnharmND prints the singular values if you put PrintSingularVals=yes
in your input �le. For each equation there is an associated singular value
140 Appendix A: AnharmND User's Manual
indicating how `important' it is. A relatively small singular value relates to
an equation that does not change the quality of the �t very much, but does
change the values of the �tted coeÆcients. Before computing the values of the
coeÆcients from the decomposed A, the program sets �i for which �i=�0 < �
to 0. This implies that the associated equations will not in uence the values
of x. Using the option SvdLimit = <some value> the limit for setting �i to
0 is changed. The default value is � = 1 � 10�6.The �ts can be weighted. The weight factors attributed to the input
points are computed according to wi = Ne�f �Ei . N is a normalization factor,
f is a factor provided by the user with FitWeightSlope = <some value>, and
Ei is the energy of input geometry i.
A.9.2 Control of the Lanczos procedure
The Lanczos procedure is an iterative way to solve eigenvalue problems. If
more iterations are done, more levels are found with greater accuracy. The
number of iterations can be set by the keyword LanczosIter. The default
value is 200, which is enough for some small problems, but certainly not for
larger ones. A `raw' Lanczos method produces `spurious eigenvalues', which
are not really eigenvalues, but an artifact of the method. Also, eigenvalues
are found with very poor accuracy, and they may be found more than once.
The program has several ways of �ltering these values out. The most impor-
tant way of �ltering out spurious eigenvalues is computing the deviation of
the expectation value of the corresponding eigenvector:phE2i � hEi2. If the
eigenvector is exact, it should have a deviation of 0. The program has two pa-
rameters to decide whether the eigenvalue is accurate enough (`sharp enough')
or not: LancMaxAbsDelta and LancMaxRelDelta. If the absolute value of the
eigenvalue is smaller than LancMaxAbsDelta, twice the deviation (i.e., the
width) divided by the absolute value of the eigenvalue should not exceed
LancMaxRelDelta. In the other cases twice the deviation should not exceed
LancMaxAbsDelta. Default values are:
LancMaxAbsDelta = 1.0e-5
LancMaxrelDelta = 1.0e-3
These keywords assume atomic units always. If a system has degenerate levels,
the program tries to �nd a complete set of orthogonal wave functions for each
degenerate level, but no more. If the program �nds two eigenfunctions that
have the same eigenvalue within the limits of LancMaxAbsDelta, it computes
the overlap between the two normalized vectors. If this overlap is smaller
than 1�DegenerateMinNorm, a Schmit-orthogonalization is done, which thenyields two orthogonal vectors. If more wave functions are found with the same
eigenvalue, it is attempted to construct a larger orthogonal set. By default
we have DegenerateMinNorm = 0.02.
Adjusting computational parameters 141
If the memory usage of the program becomes too large, the vectors pro-
duced by the Lanczos method can be written to disk. The following options
apply:
LanczosDisk = yes # as opposed to default `no'
LanczosDir = /TMP # directory for putting temporary file
LanczosBufLen = 100 # default value
LanczosTransBufLen = 100 # default value
The �les AnharmND creates for temporary storage of vectors are named
VecStorFile<pid> <nr>, where <pid> is the process id of the running job,
and <nr> is a number starting at 1, which will raise if more �les are used.
LanczosDir defaults to the current directory. The keywords LanczosBufLen
and LanczosTransBufLen control how many vectors are kept in core memory
during the Lanczos procedure and the transposing of the matrix on disk which
is necessary after the Lanczos procedure. Larger values for these bu�ers mean
larger memory usage, but also higher speed.
Normally the lanczos procedure will start with a random starting vector.
It is possible to choose a starting vector in the following way:
LanczosStartVec = 1 0 0 1
This results in a starting vector with equal, non-zero coeÆcients for the �rst
order Hermite function in the �rst and in the fourth coordinate, and zero
coeÆcient for all others. The idea is that the lanczos procedure is started
with a certain bias towards eigenfunctions that are similar to the starting
vector, to ensure they will be found within a reasonably small amount of
iterations. This doesn't seem to work very well in practice.
For larger problems where the eigenvalues are very closely spaced, the
Lanczos procedure doesn't always �nd the desired eigenfunctions within a
reasonable� number of iterations. AnharmND provides the method of poly-
nomial modi�cation of the eigenvalue spectrum to improve the relative spacing
of eigenvalues. If you want to �nd all eigenvalues below a certain value Em,
and the eigenvalue spectrum runs from E0 to Ee, you have to �nd a polynomial
f(E) so thatf(Em)� f(E0)
f(Ee)� f(Em)�
Em � E0
Ee �Em:
The lanczos procedure is then performed with f(H) instead of the Hamiltonian
H. The operator f(H) has the same set of eigenvectors f ig as the originalH, but the desired vectors are found more easily, because the eigenvalues are
� The program uses the eigenvectors for a considerable part of its calcula-
tions. If only eigenvalues were required, it would be feasible to do many more
iterations.
142 Appendix A: AnharmND User's Manual
not as closely as those from H. The eigenvalues of H are then computed as
h ijHj ii.The polynomial should be speci�ed in the input �le by its coeÆcients,
leaving out the constant term, which has no e�ect anyway, as follows:y