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Volume 17 1, number 1,2 CHEMICAL PHYSICS LETTERS 27 July
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Quantum mechanical reactive scattering by a multiconfigurational
time-dependent self-consistent field (MCTDSCF) approach
Audrey Dell Hammerich, Ronnie Kosloff Department of Physical
Chemistry and The Fritz Haber Research Center for Molecular
Dynamics, The Hebrew University, Jerusalem 91904, Israel
Mark A. Ratner Department of Chemistry, Northwestern University.
Evanston, IL 60208, USA
Received 22 February 1990; in final form 9 May 1990
The major obstacle to the description of systems containing a
large number of degrees of freedom is the exponential increase of
computational time and effort with dimensionality. A strategy is
presented to overcome this obstacle as well as the shortcoming of
the omission of correlations, while still maintaining the
simplicity and strengths of a mean-field description, based upon
iden- tifying the crucial dynamical correlations and incorporating
them with multiconfigurations. The collinear reactive scattering of
H + Hz illustrates the techniques involved and their adaptability,
flexibility, and breadth of applicability. MCTDSCF simulations,
constructed from time-dependent variational principles, are
compared with the numerically exact solution of the Schrijdinger
equation, agreement is found.
1. Introduction
A variety of theoretical approaches are applied to describe,
model, and simulate basic dynamical phenomena [ l-201. Yet, there
currently exists no sufficiently accurate and feasible methodology,
shown to be valid for all regimes of dynamics, that is applicable
to systems containing a large number of degrees of freedom. The
major obstacle is exponential growth of computational time and
effort with dimensionality. In order to be potentially successful,
any treatment must therefore seek to simplify the many-body problem
by incorporating manageable approximations which neither neglect
nor obscure the relevant chemistry and physics.
In many areas, mean-field approximations are proven procedures [
21-251 for overcoming the scaling of effort with dimensionality.
Their key attribute is the reduction of dimensionality by
decomposing one D-di- mensional problem into D one-dimensional
problems. This is amply demonstrated in molecular dynamics by
applications of the self-consistent field (SCF) approximation [
26,271. However, simple mean-field theories, static SCF, are not
broadly applicable to the whole range of molecular dynamics
phenomena. In addition to neglecting all correlations amongst the
various degrees of freedom, SCF methods have coordinate system rep-
resentations which are energy dependent, and their static nature
precludes simulating dynamical processes.
The time-dependent analogue to quantum SCF, the time-dependent
self-consistent field (TDSCF) method [ 28 1, unlike static SCF, is
applicable to dynamic processes and addresses part of the
correlations by incor- porating SCF in an explicitly time-dependent
formulation. TDSCF introduces an effective time-dependent po-
tential under the influence of which energy may be transferred
between different modes of the system, yielding
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time correlations between these modes. Until somewhat recently,
TDSCF was emphasized more in nuclear than molecular problems [ 291
and while encouraging, its applicability in understanding molecular
dynamics is still in its infancy [ 30-391. Though the approach can
indeed treat a broader range of problems, it nevertheless is an
inadequate theoretical tool for simulating many dynamical systems
of interest. Most notably, it can omit important correlations.
Without such correlations, TDSCF methods are generally unable to
correctly describe situations when the wavefunction bifurcates into
two or more parts, as in reactive scattering and nonadiabatic
transitions.
2. MCTDSCF approach
2.1. Objectives
A simple way to account for the important omitted correlations
in TDSCF is to systematically correct by the addition of
configurations, producing a multiconfigurational time-dependent
self-consistent field (MCTDSCF) description [ 34,35,38,39]. This
procedure does not increase the dimensionality of the problem and
the computational effort does not depend too critically upon the
number of configurations. The main idea behind the
multiconfiguration improvement is to produce a more flexible
description by incorporating the physically relevant correlations.
While MCTDSCF is a very promising and intuitively appealing
approach, its possibilities have barely been explored [
34,35,38-401,
The principal impetus to a multiconfigurational treatment is to
give more flexibility to the wavefunction describing the system
while at the same time rendering the general many-body problem
tractable. The main assumption of the simple TDSCF approach is the
decomposition of the one D-dimensional problem into D
one-dimensional ones by appealing to a mean-field treatment where
the total wavefunction is represented by a Hartree product of D
single “particle” wavefunctions. Thus, rather than attempt to solve
the time-dependent Schriidinger equation for the often intractable
one D-dimensional equation of motion of the exact Y, the wave-
function is approximated (via TDSCF) as the product
!P( I, 2, . ..) QI)=,&NjJ) *
In the multiconfigurational approach correlations are introduced
into the right-hand side of eq. ( 1) by in- cluding more than one
Hartree product in the wavefunction. For each configuration, the D
one-dimensional equations of motion are then variationally solved
for the $(j, t) employing the MCTDSCF approximation for
ul
where M configurations are included in eq. (2 ).
2.2. Mathematical formulation
Assuming that the initial state of the system described by the
multiconfigurational wavefunction given in eq. (2) is fixed and
known, its time evolution can be ascertained by appealing to the
McLachlan variational principle [ 4 1 ] which requires that the
functional 1, where
I= J H . . . ifig-l?!Pr(ifig-H!P)dl d2...dD, --oo (3)
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be stationary with respect to arbitrary variations of !i?
Minimization of the space integral leads to a set of equa- tions of
motion for the @,(J t). Without additional information this set is
indeterminate. The nature of the conditions or constraints placed
upon the wavefunction and how they are implemented as the system
evolves in time are determined by the physically relevant
correlations being introduced.
A condition arises naturally when the state of the system is
identified at its asymptotic limits: a configuration for reactants
and another orthogonal configuration for products. In order to
generalize the idea of orthogonal configurations, one spatial
dimension is chosen on which orthogonality is to be invoked. For
clarity, denote this spatial dimension as D and the wavefunction on
this space as x,,, (D, f ). Then the wavefunction of eq. (2 ) can
be equivalently written as
(4)
Imposing normalization on the D- 1 spaces containing the 9, and
orthogonality on the Dth space
(9mCi,~)lbnti,t))=l, (xmU4f)lxn(D,t)>=O, m#n,
along with total normalization for the wavefunction, leads
to
(5)
(6)
(Note that the above normalization conditions are introduced
purely for notational simplicity.) This choice for the wavefunction
is related to a projection operator P which determines the
correlations in the Dth space. This operator is a sum of elementary
orthogonal projectors
klxn = &2ni^x, 3 (7) which effects a partitioning of the Dth
space into the direct sum of M subspaces. For greater generality,
the Mth elementary projector is defined on the complementary
subspace of the other M- 1 projectors
M-l
P,=f- c Pm. (8) PI=,
Hence the amplitudes ( ,ym Ix,> of eq. (6) are the
probabilities associated with the projectors p,. With the desired
correlations included in the state description and with a
projection operator defined which
determines these correlations, one can return to the SchrGdinger
equation and its variational solution. There is great latitude in
the manner by which MCTDSCF equations of motion can be derived. Not
only is there a choice in the space upon which the projection is
carried out and the specific projection to be used, there is also
flexibility in implementation of the projection. In particular, two
cases can here be identified which differ ac- cording to how and
when the projection is accomplished.
In the first case, the equations of motion for the projected
wavefunctions are directly obtained. Using the previously defined
projection operator, the time-dependent Schriidinger equation can
be written as
where ti= f is the sum of projections, which reduces to the M
equations in terms of the elementary projectors
For each of the above equations a functional can be defined. The
resulting variational solution yields the MCTDSCF equations of
motion for the multicontigurational wavefunction given in eq. (4)
assuming that en-
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ergy is conserved via the x motion. Thus for the nth
configuration, its D one-dimensional equations of motion are
(lib)
Analogous equations for the projected motion exist for the other
configurations. In the second case, the constraints are directly
applied to the functional of eq. (3). The equations of motion
so derived are identical for the @ degrees of freedom. Defining
a matrix of overlaps
the equations of motion for the x degrees of freedom are
D-l
-ifi c (hn(k t) I&(k t) > P.di t) xm(D, t) , k=l >
(12)
(13)
where the prime denotes omission of the kth term from the
product in eq. ( 12 ). As the evolution in the equa- tions for the
x degrees of freedom is unprojected, the projections are performed
after integrating the equations of motion. Though this latter case
is not variationally equivalent to explicitly employing a
time-dependent pro- jection operator, it does provide a simple
procedure for evolving the projection in time. Applications of both
cases (projection prior to integrating the equations of motion and
post-integration projection) will be illustrated.
3. Numerical results and discussion
For an initial assessment of the viability of an MCTDSCF
approximation, a physical system is chosen where only two
configurations in eq. ( 1) need be considered. The process examined
is the collinear scattering of a hydrogen molecule by molecular
hydrogen. This system has been exhaustively investigated and has a
reliable potential energy surface (ref. [ 42 1, with
parametrization of ref. [ 431). Furthermore it displays purely
quantum effects. The ability of a formalism to incorporate the
relevant correlations in such a scattering problem is a stringent
test of the approximation and is a precursor to extension of the
methodology to higher dimensions and multisurfaces.
The multiconfigurational approach affords great freedom in the
choice of Hamiltonians, wavefunctions, ini- tial conditions,
projections, and in implementation. In order to explore this
flexibility, the two methods for implementation introduced in
section 2.2, denoted as MCTDSCF-1 and MCTDSCF-2 respectively, were
em- ployed in reactive scattering simulations at several
representative collision energies. These simulations are com- pared
with both the numerically exact propagation obtained via a
Chebychev expansion of the evolution op- erator [ 441 and with the
single configuration simple TDSCF approximation.
The calculations were performed using bond coordinates. In these
coordinates coupling between modes is found in the kinetic energy
operator as well as in the potential operator. Computation time
would be saved
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if the potential could be written as a sum of products of
potential terms of the different modes. The LSTH potential [ 42,431
used in this work does not have this form and as a result the
potential evaluation became the most time consuming part of the
calculations. In all calculations spatial derivatives were
calculated using the fast Fourier transform method [ 201. For
conciseness, only simulations and data with the initial coniditions
summarized in table 1 are reported.
The equations of motion were integrated by a scheme which
employs a low order polynomial approximation to the evolution
operator [ 45 1. This scheme has the very attractive feature of
allowing for variable time steps during the propagation.
Considerable time savings can result from using this option as the
largest eigenvalue determines the upper limit to the time step in a
fixed time step propagation such as second-order differencing
(SOD). In contrast, the time step of a variable method can be large
in asymptotic regions where the potential variations are small and
small in the interaction region where the potential varies
significantly. This method has been shown to be equivalent to the
short iterative Lanczos procedure, demonstrated to offer an
accurate and flexible alternative to other existing schemes for
propagating the time-dependent SchrSdinger equation
[461. For this scattering application, the polynomial
approximation was not only an alternative propagation method,
it proved to be an imperative choice for stability as SOD does
nol work. The difficulty stems from the SOD requirement of
calculating matrix elements between successive time intervals [
20,46,47], for example
(fi)=(Y(&At)l@Y(t)), (14)
and the multiconfigurational self-consistent Hamiltonian. These
two conditions result in complex expectation values, even for
observables!
Fig. 1 portrays significant time frames of the evolving dynamics
for the highest collision energy reported. The leftmost panel
displays the numerically exact simulation and serves as the
reference by which the ap- proximate methods can be assessed.
Starting at 0 au with the initial state on the right, denoted by
the lighter contour lines, the wavepacket spreads while traversing
the entrance channel. The frame at 800 au evidences a slight
extension of the y bond coordinate over its initial value. The
initial state is not an eigenstate of the Hamiltonian, not even
asymptotically. Hence the wavepacket oscillates, undergoing small
excursions between the inner and outer vibrational potential walls
with a period determined by the initial “translational” mo- mentum,
as it moves down the entrance channel. Upon entering the
interaction region at 1400 au the wave- function collides with the
repulsive wall of the potential creating interference between
incoming and reflected parts. Subsequent motion is consistent with
the classical bobsled effect [ 16,481 giving rise to an oscillatory
pattern for the reactively scattered part of the wavefunction. Some
tunneling is observed at late times when
Table I Propagation parameters
( 1 )initial state Morse oscillator eigenfimction in y
coordinate
0.99x, 0.99 v=Q 0.01 v= 1
0.01x2 1.00 v=2
Gaussian wavepacket in 1 coordinate centered at 9.0 with width
0.5 and initial momenta, corresponding to the three collision
energies simulated, of
0,: -6.0, -5.0, -4.0 h: -5.9, -4.9, -3.9
(2) potential: LSTH potential surface [42,43] truncated at
dissociation limit: &=a.1 74474
(3) grid: dx=dy=O.15 in bond coordinates
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EXACT
1600
MCTDSCF-1 MCTDSCF-2
27 July 1990
TDSCF
i/ / j / ! 1 I/ 1 pL . ^_ L . . _
~-~~~~:~~~~~::~:~~~~:~~~~~~~~~~~ .-.._........_. -
.._.............. ._.._........_.. - ..___....................
Fig. 1. A comparison of scattering simulations derived from the
numerically exact result with those from the multiconfiguration
pre- (MCTDSCF-1) and post- (MCTDSCF-2) integration projection
approaches and from the single configuration (TDSCF) approxima-
tion at a total energy of -0.1440 au. The bond coordinate axes are
all 11.1 au except for the 15.6 au axes at time frame 3100.
the reaction has almost terminated. Note the suggestion of a
resonance state at 3 100 au where a small portion of the
wavefunction resides along the symmetric stretch coordinate. About
one-third of its amplitude has pen- etrated into the classically
forbidden potential region with energy in excess of the average
total energy.
Comparing the exact result to both the single configuration and
multiconfiguration propagations, one is struck by the total
inability of the simple TDSCF approximation to yield any reacted
products. Apparently, corre-
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lations introduced by the time-dependent potential at the single
configuration level are an ineffective mech- anism for transferring
energy between the modes and the contracted state is instead
reflected off the inner and outer potential walls about the turning
point. It is well known that SCF methods work best for coordinate
sys- tems specifically chosen so that the modes are as uncorrelated
and separable (in either the spatial or frequency domain) as
possible. This physical separability restriction is generally eased
in TDSCF methods by the time dependence of the mean fields. In the
case of a scattering event the more important correlations are
dynamic due to the changes in interatomic distances which naturally
accompany the reaction. The last panel of fig. 1 shows that the
time dependence of the TDSCF mean fields is unable to represent the
essential features of these correlations so that the energy
exchanged between the modes is insufficient for reaction. This
failure has been also observed in Jacobi coordinates in which the
only coupling between the two modes comes from the potential terms.
Nevertheless, the short time dynamics are well reproduced. Only at
1200 au when the exact wave- function begins to sample the reactive
channel does the overlap of the TDSCF state with the exact state
start to deviate from a value of near unity (fig. 2). This is
perfectly in accord with the majority of TDSCF inves- tigations
which are generally successful when applied to basically single
channel dynamics as in the various studies of such “half reactions”
as predissociation and photodissociation.
The results for the two multiconfiguration approximations given
in the middle two panels of fig. 1 are in stark contrast to the
simple TDSCF approximation. Whether the projection is performed
prior to integrating the equations of motion (MCTDSCF- 1) or as a
post-integration procedure (MCTDSCF-2), the consideration of only
two configurations captures the essential aspects of the relevant
correlations between the modes al- lowing for the energy transfer
necessary to describe and simulate this scattering event. The
entrance channel dynamics, where a correctly more contracted
wavefunction is evolving, exhibit better agreement with the exact
result than does the TDSCF description. While a more quantitative
comparison of the MCTDSCF-1 and -2 methods will be given later,
clearly the post-integration projection displays dynamics which are
qualitatively in better agreement with the exact dynamics. Even
such fine details as the wavefunction skewing at 800 au and the
greater extent of delocalization of the reactively scattered as
well as backscattered portions of the wave- function are well
reproduced. However, there do exist differences between the exact
and MCTDSCF propa- gations. In particular, the bimodal distribution
of that portion of the wavefunction in the reactive exit channel is
absent. With only two configurations, it may be expecting too much
of an approximate theory to be able to
. \ _ ;0.1440 SC__
0.40 I I I I Fig. 2. Overlaps of the exact propagation with the
post-integra- 0 850 1700 2550 3400 tion projection approximation
(MCTDSCF-2). The dashed line
represents the modulus of the overlap and the solid line the
over- time lap of the mod&.
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Table 2 Branching ratios
Energy EXACT MCTDSCF-1 MCTDSCF-2 TDSCF
-0.1549 23.8 0 17.6 -0.1500 53.5 52.9 39.4 -0.1440 75.0 84.5
75.5 0
represent bifurcation into two distinct channels as well as an
additional correlated separation. The multiconfigurational
approaches were also examined at two lower collision energies and
compared with
the exact propagation. The post-integration projected MCTDSCF
approximation again showed the better qual- itative agreement. At
the lowest collision energy, where tunneling appears to be an
important mechanism, the preprojection procedure (MCTDSCF-1 ) gave
no reactively scattered products. Yet it showed better quanti-
tative agreement at an intermediate energy with respect to the
branching ratio. Table 2 summarizes the branch- ing ratios obtained
at the three different collision energies. The high result for
MCTDSCF- 1 is consistent with the greater percentage of its
wavefunction which reaches the saddle point before collision with
the repulsive wall. The incoming part of the wave interferes with
the part reflected off the potential, positioning amplitude in the
saddle region into the exit channel. Unlike the exact and MCTDSCF-2
results, here the wavefunction is not rather sharply peaked but
displays high probability throughout a large region of
configuration space. The dynamics of the MCTDSCF-2 wavefunction at
the intermediate energy which gives a lower than actual branch- ing
ratio behaves in an opposite manner to that given above. However,
there is an important difference. Qual-
Fig. 3. “Quantum” trajectory representations of the scattering
simulations. The heavy solid line is for theexact propagation while
the dotted and dashed lines denote the MCTDSCF-1 and MCTDSCF-2
approximations respectively. The light solid line, which is
displaced by 0.25 au, is a nonreactive case, TDSCF at high and
MCTDSCF-1 at low energies. x average
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itatively the approximate state is in excellent agreement with
the exact wavefunction as far as spatial extent and even the
position of peak maxima and other fine details are concerned. The
lack of agreement is solely in intensity. At later times, the
approximations at lower energies may also be suffering from an
inability to properly represent tunneling. The transfer of
amplitude from one configuration to another is governed by the
specific choice of projection, including its time dependence, and
by the overlap of 9, and &. While the addition of more
configurations will always improve the approximations, the
incorporation of another configuration on the x spatial dimension
should definitely be considered.
More insight into the nature of the approximations can be gained
by inspecting other average properties. One way to visualize the
quantum molecular dynamics is to average the bond distances and
form a “quantum” trajectory. Fig. 3 represents the resultant
trajectories for the scattering simulations in fig. 1. For all of
the ap- proximations the classical turning point is well estimated,
exhibiting the expected decrease with increasing en- ergy, and the
entrance channel dynamics are well reproduced.
The success of the multiconfiguration approach depends on the
ability of the different configurations to cap- ture the essence of
the dynamical behavior. Fig. 4 displays examples of the two
approximations, MCTDSCF- 1 and MCTDSCF-2, to the dynamical
correlation. By construction the wavefunctions 1, and ,Q are
orthogonal
MCTDSCF-1
Re(dhxl + +2x2)
MCTDSCF-2
Fig. 4. Decomposition of the multiconfiguration wavefunctions
into their respective configurations for the -0.1440 au simulations
at 2300 au. Shown are the real parts ofeach separate configuration
and their sum. The MCTDSCF-1 wavefunction is constructively formed
from spatially separated configurations while the post-integration
projected MCTDSCF-2 wavefunction exhibits a great mixing of its
constituent configurations whose orthogonality is maintained by
phase mismatch.
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(eqs. (4) and (5 ) ). The manner in which this orthogonality is
constructed is a direct indication of the cor- relations
incorporated by the methods. The dominant dynamical correlation in
reactive scattering is the split- ting of the wavefunction into
reactive and nonreactive parts. The MCTDSCF-1 method constructs a
spatial orthogonality (fig. 4 left panels) which addresses the
observation that the two chemical channels are spatially separated
from each other. On the other hand in the MCTDSCF-2 method the
orthogonality is constructed through phase relations (fig. 4 right
panels) which means a momentum-type correlation in which the
chemical channels are identified through the direction of the
outgoing velocity. As the momentum changes sign during the
collision the projection operator which is associated with the
correlation has to be time dependent. The probability for reaction
of the two methods compared to the exact value is presented in
table 2.
4. Conclusions
Quantum mechanical treatment of reactive scattering poses
difficult challenges for approximate theories. The combination of a
continuum spectrum in both reactant and product channels with
differing asymptotic co- ordinates are the important obstacles.
These difficulties plague application of mean-field methods because
of the need to describe the splitting of the wavefunction into
distinct chemical channels. This aspect of reactive scattering
requires that at least one configuration be included for each
differing chemical channel. The lack of configurational flexibility
is the reason for the failure of simple (single configuration)
TDSCF to describe a typical bimolecular reaction.
Multiconfiguration TDSCF is an appropriate generalization of
simple TDSCF to deal with situations with several reactive
channels. Formally, one can construct MCTDSCF equations from
time-dependent variational principles. One then has the possibility
of selecting as many configurations as required, and of
implementing the multiconfiguration treatment in a variety of ways.
It is appropriate to select physically motivated projection
operators to describe the differing configurations. Different
projections describe different correlations, and cer- tain
projeclions will clearly be more appropriate than others; the more
appropriate projections will lead to a more rapid convergence of
the description as the number of configurations is increased.
Considering that the wavefunction evolves in time from the
reactant through the collision region to the final state (a
superposition of backscattering and product channels), the
physically more reasonable choice for a projection operator is a
time-dependent projector, which separates the outgoing reacted
wavepacket from the backscattered component. Such a dynamical
projection operator has been implemented and applied to the H + H2
reaction. The resulting calculations, here denoted as MCTDSCF-2,
have better qualitative and quantitative features than either the
simple TDSCF or static projections, at least at this simple
two-configuration level.
The methods described here are intended to be used in large and
more complex dynamical encounters. It is therefore important to
examine the scaling of the method with dimensionality. Eq. (4)
suggests that the method scales as M+ 1 where M is the number of
configurations multiplied by D- 1 where D is the number of degrees
of freedom of the system. This growth in computation is slow
compared to the full dynamics which scale as the power of D. An
alternative very promising MCTDSCF approach has been formulated by
Meyer, Manthe, and Cederbaum [ 401. Their formulation scales as D2;
in their scheme more correlations are taken into account, the cost
of more elaborate computational effort. In higher dimensional
calculations, a product form for the potential is a time saving
necessity for computational feasibility. Bond coordinates with pair
po- tential construct an efficient representation.
Examination of figs. l-4 shows that the simple
multiconfiguration methods proposed here qualitatively de- scribe
the nature of the H + Hz reactive collision. Scattering of the
packet and its self interference near the sad- dle point, the
multi-peaked product density, the closer approach to the turning
point at higher energies, and the overall reactivity patterns are
appropriately described even by the very simple two-configuration
model employed here. These features are not reproduced in simple
TDSCF (again in bond coordinates as well as Jacobi coordinates),
which does not develop any component in the reactive channel.
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A number of important technical points can be made. In
particular we note that second-order differencing schemes for the
time propagation, which can be useful for exact dynamics, become
problematic in the MCTDSCF context due to complex valued averages,
that can lead to divergence. The multicontigurations and the mixed
potential and kinetic terms make the use of the split operator
propagation scheme very difficult. Alternative integration schemes
based on a low order polynomial approximation of the evolution
operator, as employed here, have a considerable advantage over
other propagation methods.
Perhaps most important, the use of time-dependent projectors to
define an MCTDSCF approximation to chemicalIy reactive scattering
seems to overcome the two difficulties that simple TDSCF has with
such systems; firstly it permits description of many channel
scattering, as will be necessary for any true reactive process with
quantum yield less than unity. Secondly, it permits physically
based definition, on the basis of a time-dependent projector, of
precisely which channels will be used to construct the
configurations. Thirdly, it contains dy- namical corrections absent
in simple TDSCF. Finally, insight is gained by watching the
dynamical evolution of the wavefunction, something that
time-independent formalisms simply cannot do.
MCTDSCF methods should share many of the advantages of simple
TDSCF (conservation of norm and energy, general applicability to
weak and strong coupling limits, classical and semiclassical
limits, good ac- curacy) and like TDSCF methods possess coordinate
dependence. The relatively slow growth of computational effort with
increase of the system’s dimensionality is a great advantage of
these methods, indicating consid- erable promise for their use in
dynamical calculations of quantum mechanical encounters.
Acknowledgement
This research was supported by a grant from the GIF, the
German-Israeli Foundation for Scientific Research and Development.
The Fritz Haber Research Center for Molecular Dynamics is supported
by the Minerva Gesellschaft ftir die Forschung, GmbH Munich,
Federal Republic of Germany. ADH gratefully acknowledges receipt of
a Lady Davis postdoctoral fellowship. MR thanks the Chemical
Division of the NSF for partial support.
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