1 MODELING MATTER AT NANOSCALES 6. The theory of molecular orbitals for the description of nanosystems (part II) 6.05. Variational methods for dealing.

Luis A. MonteroUniversidad de La Habana, Cuba, 2012

1

MODELING MATTER AT NANOSCALES

6. The theory of molecular orbitals for the description of nanosystems (part II)6.05. Variational methods for dealing with correlation energy

Optimizing the Hartree - Fock wave function

Being Fs the multielectronic Hartree-Fock wave function of any s state corresponding to a molecule or other nanoscopic system it can always be expressed as a linear combination of associated wave functions, following the algebraic formalism:

I

IsIs a

where the sum is over a sufficiently large I series of YI multielectronic reference states and asI are their corresponding participation coefficients in such s state.


Being Fs the multielectronic Hartree-Fock wave function of any s state corresponding to a molecule or other nanoscopic system it can always be expressed as a linear combination of associated wave functions, following the algebraic formalism:

I

IsIs a

where the sum is over a sufficiently large I series of YI multielectronic reference states and asI are their corresponding participation coefficients in such s state.

Y must be an orthonormal basis.


Obviously, asI participation coefficients could be optimized on the grounds of a variational approach to obtain the I series giving a s state of minimal total energy.


Obviously, asI participation coefficients could be optimized on the grounds of a variational approach to obtain the I series giving a s state of minimal total energy.

Therefore, the resulting Fs for each state of the system will better approach the exact wave function as the total energy becomes smaller.


If multielectronic reference functions are antisymmetrized products or Slater determinants resulting from a Hartree – Fock calculation, a typical electronic configuration is that where each spin orbital represents a given existing electron.

where yi reference functions are spin orbitals describing the state of an electron of rnan ≡tn spatial and spin coordinates.

)()(det)!(

)(...)()(

......

)(...)()(

)(...)()(

)!(

111

21

22212

12111

1

nini

NNNN

N

N

I

rrN

rrr

rrr

rrr

N


)(...)()(

......

)(...)()(

)(...)()(

)!(

662616

622212

612111

10

rrr

rrr

rrr

N

)()()()()()(

)()()()()()(

)()()()()()(

)()()()()()(

)()()()()()(

)()()()()()(

)!(

610510410310210110

665646362616

655545352515

635343332313

625242322212

615141312111

1

rrrrrr

rrrrrr

rrrrrr

rrrrrr

rrrrrr

rrrrrr

NI

Distribution of different spin orbital populations can vary from that of the ground state Y0. Each one of the I electron populations could be a valid basis representation of the system serving for a variational optimization.


Each determinant showing a different electron distribution to that of the ground state is known as an excited configuration.

)()()()()()(

)()()()()()(

)()()()()()(

)()()()()()(

)()()()()()(

)()()()()()(

)!(

610510410310210110

665646362616

655545352515

635343332313

625242322212

615141312111

1

rrrrrr

rrrrrr

rrrrrr

rrrrrr

rrrrrr

rrrrrr

NI


Configurations could correspond to one of the following types:LUMO+3

LUMO+2

LUMO+1

LUMO

HOMO

HOMO-1

HOMO-2

HF S S D D T Q

HF: ground state; S: monoexcited configuration; D: double excited configuration; T: triple excited configuration; Q: quadruple excited configuration


Departing from a solution of the Hartree – Fock multielectronic wave function a very large number of configurations can be made depending on the number of electrons entering different occupations of spin orbitals:

where means a p order excited configuration, redistributing p charges among all initially unoccupied spin orbitals.

)0(0

...)4()4()3()3()2()2()1()1()0(0

)0(0

IIsI

IIsI

IIsI

IIsI

IIsIs

aaaaa

a

)(0

)(0

ppa


Therefore, the solution of a certain n electron system under the Hartree – Fock procedure represented by N spin orbital basis functions will amount m = N – n non occupied spin orbitals.

It can be shown that the number of possible reference functions for this development is huge and can be calculated according to:

!!

!

!!

!

1 mn

mn

nmnn

mn

n

mn

p

m

p

nn

p

Water as an example

If we develop a minimal Slater basis function for water:

We will have n = 10 electrons and N = 14 possible basis spin orbitals when the Hartree – Fock molecular problem was solved. Therefore m = 4 orbitals will remain as non occupied.

H: 1sa1, 1sbH: 1sa1, 1sbO: 1sa1, 1sb1, 2sa1, 2sb1, 2pxa1, 2pxb1, 2pya1, 2pyb1, 2pz , a 2pzb

Water as an example

If we develop a minimal Slater basis function for water:

We will have n = 10 electrons and N = 14 possible basis spin orbitals when the Hartree – Fock molecular problem was solved. Therefore m = 4 orbitals will remain as non occupied.

H: 1sa1, 1sbH: 1sa1, 1sbO: 1sa1, 1sb1, 2sa1, 2sb1, 2pxa1, 2pxb1, 2pya1, 2pyb1, 2pz , a 2pzb

Therefore, the number of possible electronic configurations will be:

1001

!4!10

!14

!!

! mn

mn

Water as an exampleIf the basis function is as simple as 3-21G, basis atomic orbitals would be:

We will again have n = 10 electrons but N = 26 possible basis spin orbitals when the Hartree – Fock molecular problem was solved. Therefore m = 16 orbitals will remain as non occupied.

H: 1sa1, 1s , b 1sa’, 1sb’H: 1sa1, 1s , b 1sa’, 1sb’O: 1sa1, 1sb1, 2sa1, 2sb1, 2pxa1, 2pxb1, 2pya1, 2pyb1, 2pz , a 2pz , b 2sa’,

2sb’, 2pxa’, 2pxb’, 2pya’, 2pyb’, 2pza’, 2pzb’

Water as an exampleIf the basis function is as simple as 3-21G, basis atomic orbitals would be:

We will again have n = 10 electrons but N = 26 possible basis spin orbitals when the Hartree – Fock molecular problem was solved. Therefore m = 16 orbitals will remain as non occupied.

Therefore, the number of possible electronic configurations will be much higher:

H: 1sa1, 1s , b 1sa’, 1sb’H: 1sa1, 1s , b 1sa’, 1sb’O: 1sa1, 1sb1, 2sa1, 2sb1, 2pxa1, 2pxb1, 2pya1, 2pyb1, 2pz , a 2pz , b 2sa’,

2sb’, 2pxa’, 2pxb’, 2pya’, 2pyb’, 2pza’, 2pzb’

5311735

!16!10

!26

!!

! mn

mn

Correlation energy

Being the Hartree – Fock equation of eigenvalues and eigenfunctions of a given I electronic configuration YI :

A certain matrix element on the ground of atomic orbitals can be written in a simplified way as:

where

III EF ˆ

i

Ii

Ii

I ccHF ˆ

211 21212112

ddr

Correlation energy

The total energy of such I configuration then remains as:

being a matrix element of the density matrix in the I state:

on the grounds of cmi molecular orbital coefficients for atomic orbitals cm and cn.

core

IIIHFI VHE

2

1ˆ

i

Ii

Ii

I cc

Correlation energy

On the other hand, the “exact” expression for the s state energy Es, as an output of optimizing linear combinations of all reference I states:

would be expected to become from:

IIsIs a

sss E ̂

Correlation energy

It is evident that finding the Fs wave functions and their corresponding eigenvalues Es after optimizing Hartree – Fock electron configurations means that we are trying to attain a form of correlation energy, given the corresponding definition, as in the case of a ground state:

HFtotcorr EEE 0

Correlation energy

It is evident that finding the Fs wave functions and their corresponding eigenvalues Es after optimizing Hartree – Fock electron configurations means that we are trying to attain a form of correlation energy, given the corresponding definition, as in the case of a ground state:

HFtotcorr EEE 0

It is congruent with the definition of correlation energy if considering the variationally optimized energy as “exact”.

The method of configuration interaction (CI)

The method of configuration interaction (CI) means the variational optimization of multielectronic states on the basis of linear combinations of several different configurations of the given system. Such configurations are expressed as their respective determinants being built from a previous one electron wave function procedure.


The method of configuration interaction (CI) means the variational optimization of multielectronic states on the basis of linear combinations of several different configurations of the given system. Such configurations are expressed as their respective determinants being built from a previous one electron wave function procedure.

If the expansion includes all possible configurations, then this is called as a full configuration interaction procedure which is said to exactly solve the electronic Schrödinger equation within the space spanned by the one-particle basis set.


The matrix representation to solve the problem is:HA = EA

It means the evaluation and diagonalization of the HIJ matrix elements in:

NNN

N

N

HH

HHH

HHH

....

......

...

...

0

11110

00100

H


The matrix representation to solve the problem is:HA = EA

It means the evaluation and diagonalization of the HIJ matrix elements in:

It will output:– asI eigenvectors for the A transformation matrix– Es eigenvalues of relative energies corresponding to all s

states, resulting in the E diagonal matrix.

NNN

N

N

HH

HHH

HHH

....

......

...

...

0

11110

00100

H


The general formulation for HIJ matrix elements is*:

Depending on the energy of interaction among all and each one of the I and J configurations, taken by pairs.

NJNI

NJNIIJ

H

dHH

...ˆ...

...ˆ...

2121*

2121*

* Nesbet, R. K., Configuration interaction in orbital theories. Proc. Roy. Soc. (London) A 1955, 230 (1182 ), 312-321.


The general formulation for HIJ matrix elements is*:

Depending on the energy of interaction among all and each one of the I and J configurations, taken by pairs.

NJNI

NJNIIJ

H

dHH

...ˆ...

...ˆ...

2121*

2121*

It builds a symmetrical matrix.

YIJ basis functions are Slater determinants or any orthonormal set of multielectronic wave functions.* Nesbet, R. K., Configuration interaction in orbital theories. Proc. Roy. Soc. (London) A 1955, 230 (1182 ), 312-321.


From this general formulation, it must be noticed that:

- Each HIJ can exist for interactions between I and J excited configurations independently of the number of involved electrons.

NJNI

NJNIIJ

H

dHH

...ˆ...

...ˆ...

2121*

2121*




NJNI

NJNIIJ

H

dHH

...ˆ...

...ˆ...

2121*

2121*

- When the basis to build a configuration is orthonormal interactions between terms differing in more than two monoelectronic orbitals vanish.




NJNI

NJNIIJ

H

dHH

...ˆ...

...ˆ...

2121*

2121*

- When I = J then HII means the energy EI of such basis configuration.

- When the basis to build a configuration is orthonormal interactions between terms differing in more than two monoelectronic orbitals vanish.

Evaluation of CI matrix elements

Evaluation of HIJ elements of the CI’s H matrix remains as the key problem, as occurs in all variational quantum calculations.


Evaluation of HIJ elements of the CI’s H matrix remains as the key problem, as occurs in all variational quantum calculations.

It must be taken into account that basis functions are no longer one electron wave functions (molecular or atomic orbitals) but Hartree – Fock’s multielectronic configurations after the previous optimization of the initial occupation, that usually is the ground state:

I

IsIs a

Evaluation of CI matrix elementsHIJ terms depend on projections of a given I configuration or multielectronic state on another one denoted as J. It means an scalar product in the configuration space.

JIJI d **

Evaluation of CI matrix elementsHIJ terms depend on projections of a given I configuration or multielectronic state on another one denoted as J. It means an scalar product in the configuration space.

As each configuration is expressed as a determinant of one electron wave functions:

JIJI d **

IJ

lklk

JJII

JIJI

D

lkd

d

d

det*det*det

...det...**

11111

212

21

1

**

where orbitals of I determinant are numbered by k sub indexes and those of J by l’s.


Such determinant can be developed in terms of minors of p order, according the “p” one electron functions differencing I and J configurations.

lkDIJ | For single excitations 2121 | llkkDIJ For double excitations ppIJ lllkkkD ...|... 2121 For any “p” order excitation

Evaluation of CI matrix elementsAs one electron basis functions (spin orbitals) building Slater determinants are orthonormal, the sole none vanishing elements of DIJ determinant are those where k and l are the same, although appearing in two separate configurations.

Evaluation of CI matrix elementsAs one electron basis functions (spin orbitals) building Slater determinants are orthonormal, the sole none vanishing elements of DIJ determinant are those where k and l are the same, although appearing in two separate configurations.If spin orbital ordering of both I and J determinants are the same, non vanishing terms will appear in the diagonal of DIJ. Any occupation difference between them will also brought vanishing diagonal terms when l ≠ k.

Evaluation of CI matrix elementsAs one electron basis functions (spin orbitals) building Slater determinants are orthonormal, the sole none vanishing elements of DIJ determinant are those where k and l are the same, although appearing in two separate configurations.If spin orbital ordering of both I and J determinants are the same, non vanishing terms will appear in the diagonal of DIJ. Any occupation difference between them will also brought vanishing diagonal terms when l ≠ k.

Therefore, for all determinant minors in general:

IJppIJ lllkkkD ...|... 2121


Thus, the CI matrix element is generally expressed as:

...

...|...ˆ***!3

1

|ˆ**!2

1

|ˆ*

...ˆ...

...ˆ...

321 321

321321

21 21

2121

21213212,1321

2121212,121

111)0(

2121*

2121*

N

kkkppIJ

N

lllkllkkk

N

kk

N

llIJllkk

N

k

N

lIJlkIJ

NJNI

NJNIIJ

lllkkkDH

llkkDH

lkDHDH

H

dHH


And in the case of orthonormal spin orbitals DIJ’s become Kroneker’s deltas:

...

ˆ***!3

1

ˆ**!2

1

ˆ*

...ˆ...

...ˆ...

321

332211

321

321321

21 21

22112121

3212,1321

212,121

111)0(

2121*

2121*

N

kkklklklk

N

lllkllkkk

N

kk

N

lllklkllkk

N

k

N

lkllkIJ

NJNI

NJNIIJ

H

H

HDH

H

dHH

cancelling most terms of all sums.


If in the case of interactions between configurations where i and j are initially filled spin orbitals transferring electrons to certain a and b initially empty spin orbitals, the calculation of matrix elements only involve the populated and unpopulated spin orbitals by:

for double excitations. Similar terms can be developed for triples and quadruples.

in the case of single excitations and

jajjijaji

aiiaIJ HH

21212121

111ˆ

21212121, abjibajijbia

IJH


Single electron excitations from the ground state can be expressed as:

leading to the matrix element between two spin orbitals of the same monoelectronic solution.

11

21212121

110

ˆ

ˆ

ai

jajjijaji

aiiaI

F

hH


Being both spin orbitals eigenfunctions of the Fock operator:

because all yi and ya orbitals are orthonormal by definition. It means that:

iaa

aiaai

aaa

F

F

1111

11

ˆ

ˆ

00 iaIH


Being both spin orbitals eigenfunctions of the Fock operator:

because all yi and ya orbitals are orthonormal by definition. It means that:

iaa

aiaai

aaa

F

F

1111

11

ˆ

ˆ

00 iaIH

It is known as the Brillouin’s theorem that establish that the ground state do not intervene in optimization of singly excited states.


As single excited configurations do not participate in ground state energy optimizations by CI procedures, such singly excited determinants only serve for energy optimizations of excited states.


As single excited configurations do not participate in ground state energy optimizations by CI procedures, such singly excited determinants only serve for energy optimizations of excited states.

Double and quadruple excited configurations are important to optimize energy of the ground state by accounting electron correlation.

Evaluation of CI matrix elementsIn the case of singly excited configuration interaction we arrived to:

Here, the term:

jajjijaji

aiiaIJ HH

21212121

111ˆ

111ˆ aiia HE

is the associated transition energy of an electron with space and spin coordinates t1 in spin orbital yi changing to occupy the formerly empty ya being defined by certain I singly excited configuration (as expressed by the corresponding Slater determinant).

Evaluation of CI matrix elementsIn the case of singly excited configuration interaction we arrived to:

Here, the term:

jajjijaji

aiiaIJ HH

21212121

111ˆ

111ˆ aiia HE

is the associated transition energy of an electron with space and spin coordinates t1 in spin orbital yi changing to occupy the formerly empty ya being defined by certain I singly excited configuration (as expressed by the corresponding Slater determinant).If we re dealing with Hartree – Fock spin orbitals:

iaia eeE


On the other hand, for the two electron terms in:

we have the general expression:

jajjijaji

aiiaIJ HH

21212121

111ˆ

as the corresponding general matrix element expressing the interaction of electrons 1 and 2 that occupied yi and yj as changing to their new occupation in ya and yb spin orbitals.

2121211

212121 12 ddabbarjibaji


By taking into account previous considerations, after some algebraic deductions*, the practical evaluation of CI singly excited matrix becomes from the following for diagonal elements:

and for off diagonal elements (being i j and a b) become:

0

2

0

211

2133

211

21211

2111

12

1212

I

airaiiaiaIIII

iaraiairaiiaiaIIII

H

eeHH

eeHH

21

121

,33

211

21211

21,11

12

12122

birajjbia

IJIJ

birajibrajjbia

IJIJ

HH

HH

* Pople, J. A., The electronic spectra of aromatic molecules. II. A theoretical treatment of excited states of alternant hydrocarbon molecules based on self-consistent molecular orbitals. Proc. Phys. Soc. London 1955, 68A, 81-9.

Some CI resultsH2O*

* Saxe, P.; Schaeffer III, H. F.; Handy, N. C., Exact solution (within a double-zeta basis set) of the Schrödinger electronic equation for water. Chem. Phys. Lett. 1981, 79 (2), 202-204.

A standard contracted Gaussian double z basis set of Dunning – Huzinaga was used with a scale factor of 1.2 for H. Geometry was previously optimized at the same level.

Some CI results

H2 equilibrium bond distance (Bohr’s)*

** Kolos, W.; Wolniewicz, L., Improved Theoretical Ground-State Energy of the Hydrogen Molecule. J. Chem. Phys. 1968, 49 (1), 404-410 after an extensive quantum calulation over all possible variables of the molecule where vibrational energy remain only 0.9 cm-1 above experimental value.

* Szabo, A.; Ostlund, N. S., Modern quantum chemistry: introduction to advanced electronic structure theory. First edition, revised ed.; McGraw-Hill: New York, 1989; p 466.

Atomic basis set SCF “Full CI” STO-3G 1.346 1.389 4-31G 1.380 1.410 6-31G** 1.385 1.396 “Exact” value** 1.401

Some CI results

H2 equilibrium bond distance (Bohr’s)*

* Szabo, A.; Ostlund, N. S., Modern quantum chemistry: introduction to advanced electronic structure theory. First edition, revised ; McGraw-Hill: New York, 1989; p 466.

Some CI results

CH2O (formaldehyde)

STO-6G 6-31G 6-311++G** CID|6-311++G**

CISD|6-311++G**

Exp.

Total energy (Hartrees)

-113.4408 -113.8084 -113.9029 -114.2245 -114.2279

rCO (Å) 1.216 1.210 1.180 1.197 1.199 1.210rCH (Å) 1.098 1.082 1.094 1.101 1.102 1.102<HCH 114.8 116.6 116.0 116.1 116.0 121.1m (debye) 1.596 3.304 2.806 2.893 2.905 2.33Relative calc. time

1 0.8 1.4 10.9 12.6

Single excitations and light

Single excitation configuration interactions (CIS) are mostly used to optimize excited state energies for simulating UV spectra of molecules.

Single excitations and light

Single excitation configuration interactions (CIS) are mostly used to optimize excited state energies for simulating UV spectra of molecules.

Therefore, it allows applications to spectroscopy, photochemistry and photonics, the science for generating, controlling and detecting photons.

Single excitations and lightAn electronic state of a given nanoscopic system can be understood as characterized by a corresponding electron density map.

Porphirin ground state charge distribution as calculated at the CNDOL/2CC||MP2|4-31G(d,p) level. Reds are negative and blues positive charges.

Single excitations and lightThe ground state shows the most stable charge cloud distribution while excited states are different, by using excitation energy to stabilize it and returning to the original ground state density map upon deactivation.

Acrolein excitation from the S0 (ground state) to S1 (first excited singlet state) as calculated at the

CNDOL/2CC||MP2|4-31G(d,p) level. Reds are negative and blues positive charges.

hn

Multiconfigurational Hartree – Fock procedures (MC-SCF)

Are there certain cases where small energy variations decide a nanoscopic procedure, as those with:• Biradicals• Non saturated transition metals• Excited states perturbing ground states• Bond breakings• Transition states in swinging transitions


Are there certain cases where small energy variations decide a nanoscopic procedure, as those with:• Biradicals• Non saturated transition metals• Excited states perturbing ground states• Bond breakings• Transition states in swinging transitions

It could be useful that interacting configurations could also be included for Hartree – Fock wave function optimizations.


Then, a more or less limited number of configurations can be chosen to be treated by independent Hartree – Fock ‘s iterative procedures, representing the active space (AS), to optimize both CI and one electron wave function coefficients.


Then, a more or less limited number of configurations can be chosen to be treated by independent Hartree – Fock ‘s iterative procedures, representing the active space (AS), to optimize both CI and one electron wave function coefficients. This procedure is called as a multiconfigurational self consistent field (MC-SCF) routine, meaning that:

is the multiconfigurational wave function and AK are the optimizing coefficients on the basis of previously optimized YK Hartree – Fock’s configurations.

K

KKMCSCF A


Each optimized YK , or configurational state function (CSF), can be:• A unique Slater determinant with a given and convenient

electron occupation of spin orbitals.• A combination of Slater determinants to correct spin

influences.


CSF’s can be expressed by orbital occupations of the initial ground state Hartree – Fock solution and their corresponding Slater determinants:

;...

)(...)()...(

......

)(...)()...(

)(...)()...(

)!(

;

)(...)()...(

......

)(...)()...(

)(...)()...(

)!(

1

2221

1111

1

1

2221

1111

1

NNNfN

Nf

Nf

L

NNNdN

Nd

Nd

K

N

N


)()( 11 i

Ki

Ki c

Each one electron spin orbital of a given K configuration is defined as the LCAO of Hartree – Fock’s method:


At the end, both coefficient sets of A CI’s and CK Hartree – Fock’s coefficient matrix for the chosen K configurations are involved in a MC-SCF optimization. They are optimized iteratively in cycles until a desired convergence limit.

)()( 11 i

Ki

Ki c



At the end, both coefficient sets of A CI’s and CK Hartree – Fock’s coefficient matrix for the chosen K configurations are involved in a MC-SCF optimization. They are optimized iteratively in cycles until a desired convergence limit.

)()( 11 i

Ki

Ki c


MC-SCF procedures are particularly demanding of computational resources.

The particular case of CAS-SCF

The complete active space self-consistent field (CAS-SCF) procedure was developed* to select an active space given by a certain NA number of electrons to participate in the variational optimization and a given M number of spin orbitals of convenience, by following certain rules.

*Roos, B. O.; Taylor, P. R.; Siegbahn, P. E. M., A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chem. Phys. 1980, 48 (2), 157-173.

.

The particular case of CAS-SCFThe total of participating orbitals are divided in three sets:

• Inactive orbitals always taken as double occupied with a total of N – NA electrons, where N is the total number of system’s electrons. They are all spin orbitals below the last reaching up to complete N – NA electrons.



• Active orbitals, giving the active space, consisting in all configurations where the chosen NA electrons can participate in M spin orbitals above the inactive space. It is recommended that the selected M orbitals were those more involved in the modeled phenomenon (i.e., p orbitals near to HOMO). Their average occupation will be a real number, between 0 and 2.



• Active orbitals, giving the active space, consisting in all configurations where the chosen NA electrons can participate in M spin orbitals above the inactive space. It is recommended that the selected M orbitals were those more involved in the modeled phenomenon (i.e., p orbitals near to HOMO). Their average occupation will be a real number, between 0 and 2.

• Virtual orbitals, always empty of electrons. They will be all those remaining above the active orbitals to reach the total number of spin orbitals of the system.

The particular case of CAS-SCFGraphical representation:

← Inactive orbitals inactivos

← Active orbitals: NA electrons and a total of M orbitals.

← Virtual orbitals

After selecting the orbitals to consider, CI is only performed on the active space, and the rest of the system is treated at a normal Hartree – Fock level.

The particular case of CAS-SCF

The example of formaldehyde:

STO-6G 6-311++G** CISD|6-311++G**

CASSCF(6,8)|6-311++G

Exp.

Total energy (Hartrees)

-113.4408 -113.9029 -114.2279 -114.0260

rCO (Å) 1.216 1.180 1.199 1.212 1.210rCH (Å) 1.098 1.094 1.102 1.094 1.102<HCH 114.8 116.0 116.0 116.1 121.1m (debye) 1.596 2.806 2.905 1.005 2.33Relative calc. time

1 1.4 12.6 313.9

CASSCF(m,n): n = number of electrons in active space; m = number of spin orbitals in active space.

1 MODELING MATTER AT NANOSCALES 6. The theory of molecular orbitals for the description of nanosystems (part II) 6.05. Variational methods for dealing.

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