The AQCC method and its application to the
calculation of excited states
Péter G. Szalay
Eötvös Loránd University
Institute of Chemistry
H-1518 Budapest, P.O.Box 32, Hungary
[email protected]
P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Plan of the talk
• Functional form of the energy and energy derivatives
• Solution of the CEPA equations
• Calculation of excited states – test results
• Calculation of transition moments – test results
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Functional form and energy derivatives[1]
Generalized Hellmann-Feynman theorem (GHF):
Suppose we have a functional of the energy
∆E = F(t; HN ),
• t: vector of the wave function parameters
P
†
• HN =
pq fpq {p̂ q̂} +
Hamiltonian)
1
4
† †
g
{p̂
q̂ ŝr̂} (second quantized
pq,rs
pqrs
P
Parameters are obtained variationally:
∂F
=0
∂tk
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for all k
2
P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Functional form and energy derivatives
The energy derivative according to a perturbation α (e.g. electric field,
nuclear displacement etc.) can be written as:
dF X ∂F dtk
∂F dHN
∂F dHN
=
+
=
,
dα
∂tk dα ∂HN dα
∂HN dα
k
first derivatives do not require the knowledge of the derivative of
parameters t:
dF X
dfpq X
dgpq,rs
=
Dpq
+
Γpq,rs
.
dα
dα
dα
pq
pqrs
Here D and Γ being one- and two-particle effective density matrices defined
by the parameters t. Note, that D and Γ do not depend on the perturbation
or on the Hamiltonian.
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P.G. Szalay: The AQCC method
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Functional form and energy derivatives
What can we do if t is not obtained variationally?
1. Take the equations defining t: Wi(t) = 0
2. Define a new functional
Pintroducing Lagrangian multipliers:
F(t, λ; HN ) = ∆E + i λiWi.
3. Apply GHF:
•
dF
dλi
dF
dti
= Wi = 0, i.e. we get equations defining t
P
dW
d∆E
= dti + j λj dtij = 0
•
This new set of equations defines λ.
4. F is now variational with respect to all parameters:
dF X ∂F dtk X ∂F dλk
∂F dHN
∂F dHN
=
+
+
=
,
dα
∂tk dα
∂λk dα
∂HN dα
∂HN dα
k
k
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P.G. Szalay: The AQCC method
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Functional form and energy derivatives
5. the energy derivative can thus be again written with the density matrices:
dF X
dfpq X
dgpq,rs
=
Dpq
+
Γpq,rs
.
dα
dα
dα
pq
pqrs
but now the effective density matrices depend on both t and λ.
The cost of the gradient calculations increases by that of the new
equation!
Methods are preferred where the parameters are determined
variationally.
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P.G. Szalay: The AQCC method
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Functional for CEPA type methods[2]
General form of the CEPA equations and energy:
hφi|HN |(φ0 +
X
cj φj )i − ciRi = 0
∆E =
X
j
cihφ0|HN |φii.
i
The obvious form of the functional:
F(c) = hφ0|HN |
X
ciφii +
i
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X
i
λi(hφi|HN |(φ0 +
X
cj φj )i − ciRi)
j
6
P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Functional for CEPA type methods[2]
General form of the CEPA equations and energy:
hφi|HN |(φ0 +
X
cj φj )i − ciRi = 0
∆E =
X
j
cihφ0|HN |φii.
i
The obvious form of the functional:
F(c) = hφ0|HN |
X
ciφii +
i
= h(φ0 +
X
ci(hφi|HN |(φ0 +
X
i
X
ciφi)|HN |(φ0 +
i
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cj φj )i − ciRi)
j
X
i
ciφi)i −
X
c2i Ri.
i
6
P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Functional for CEPA type methods[2]
General form of the CEPA equations and energy:
hφi|HN |(φ0 +
X
cj φj )i − ciRi = 0
∆E =
X
j
cihφ0|HN |φii.
i
The obvious form of the functional:
F(c) = hφ0|HN |
X
ciφii +
i
= h(φ0 +
X
ci(hφi|HN |(φ0 +
X
i
X
ciφi)|HN |(φ0 +
i
cj φj )i − ciRi)
j
X
i
ciφi)i −
X
c2i Ri.
i
If the derivative of this functional is the same as the equation defining the
method, the method is ’variational’.
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P.G. Szalay: The AQCC method
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Functional for CEPA type methods[2]
This is fulfilled if:
X
j
2 dRj
cj
dci
dRj
= 0 or
= 0 for all i.
dci
This means:
• Rj must be independent of the coefficients
• Rj can depend on the correlation energy
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Functional for CEPA type methods[2]
This condition is satisfied only for:
• ACPF[3]
• AQCC[4]
• (CEPA-v method of Pulay[5])
An equivalent form of the functional is given by:
h(φ0 +
F(c) =
where X =
P
i ci φi )|HN |(φ0
1+X
+
P
i ci φi )i
,
2 Ri
c
i i ∆E .
P
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P.G. Szalay: The AQCC method
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The AQCC/ACPF functional[3]
P
P
φ0 + h i ciφi|ĤN |φ0 + i ciφii
P
1 + G i(ci)2
F(c) =
MR-ACPF: G =
2
ne
e −2)
MR-AQCC: G = 1 − (nen−3)(n
e (ne −1)
MR-CISD: G = 1
ne: number of electrons
Stationary equations:
hφj |ĤN |
X
ciφii + (1 − G)∆Ecj
= ∆Ecj
i
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The AQCC/ACPF functional[3]
P
P
φ0 + h i ciφi|ĤN |φ0 + i ciφii
P
1 + G i(ci)2
F(c) =
MR-ACPF: G =
2
ne
e −2)
MR-AQCC: G = 1 − (nen−3)(n
e (ne −1)
MR-CISD: G = 1
ne: number of electrons
Stationary equations:
hφj |ĤN |
X
ciφii + (1 − G)∆Ecj
= ∆Ecj
i
hφj |ĤN + δij (1 − G)∆E|
X
ciφii = ∆Ecj
i
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P.G. Szalay: The AQCC method
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MR-AQCC/MR-ACPF Density[3]
One- or two-particle density matrix elements (t = {pq} or {pqrs}):
Γt =
∂(F + E0)
=a
∂Ht
CI
Γt + (1 − a)
ref
Γt
with
• a=
•
CI
•
ref
P
1+G
(ci )
iP
2
i (ci )
2
Γi: density matrix as in CI, but calculated with
MR-AQCC/MR-ACPF coefficients
Γt: density matrix based on the reference wave functions
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Solution of the CEPA equations[2]
General form of the CEPA equations:
hφi|ĤN |(φ0 +
X
cj φj )i = ciRi
j
or
hφi|ĤN |(φ0 +
X
cj φj )i + ciKi = ci∆E.
j
ˆ j i = (Ki)δij :
Introducing hφi|∆|φ
ˆ
hφi|(ĤN − ∆)|(φ
0+
X
cj φj )i = ci∆E
j
Diagonal shift form: J. L. Heully and J. P. Malrieu[6]
ˆ
This is a matrix eigenvalue equation of ĤN − ∆
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Solution of the CEPA equations[2]
General form of the CEPA equations:
hφi|ĤN |(φ0 +
X
cj φj )i = ciRi
j
or
hφi|ĤN |(φ0 +
X
cj φj )i + ciKi = ci∆E.
j
ˆ j i = (Ki)δij :
Introducing hφi|∆|φ
ˆ
hφi|(ĤN − ∆)|(φ
0+
X
cj φj )i = ci∆E
j
Diagonal shift form: J. L. Heully and J. P. Malrieu[6]
ˆ
This is a matrix eigenvalue equation of ĤN − ∆
ˆ depends on ci and/or ∆E → iterative solution required
But ∆
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Solution of the CEPA equations[2]
One can readily use a Davidson-type method:
• σ vector:
σ (i) = Hef f c(i) = HN c(i) + ∆c(i) = σ0(i) + σ1(i),
σ0: same as in CI
σ1: correction
• the small Hamiltonian matrix:
(i)
hji = c(j)σ0(i) + c(j)σ 1 = h(0)ij + c(j)∆c(i)
∆ depends on the energy → σ (i) and h change in every iterations!
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Solution of the CEPA equations[2]
However, it is enough:
• to save: σ 0 and h(0)
• recalculate c(j)∆c(i)
In case of ACPF, AQCC etc.:
the shift is the same for all configurations (∆ii = K) and therefore
hji = h(0)ji + Kc(j)c(i) = h(0)ji + Ksji,
where s is the overlap matrix in the expansion space.
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P.G. Szalay: The AQCC method
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Solution of the CEPA equations[2]
The main advantages of this procedure:
• no modification to the matrix-vector product part of the CI program
• the most expensive part of the calculation (forming the matrix-vector
product) does not need to be repeated after ∆ is updated;
• all methods can be implemented in the same fashion
• additional cost only from ∆c(i) dot-product
• excited states can be calculated
For more detail see [2]
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P.G. Szalay: The AQCC method
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Calculation of excited states[7]
Going back to the equation:
ˆ
hφi|(ĤN − ∆)|(φ
0+
X
cj φj )i = ci∆E
j
It is a matrix diagonalization
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Calculation of excited states[7]
Going back to the equation:
ˆ
hφi|(ĤN − ∆)|(φ
0+
X
cj φj )i = ci∆E
j
It is a matrix diagonalization → we can get more than one root:
ˆ α)|(φ0 +
hφi|(ĤN − ∆
X
α
α
cα
j φj )i = ci ∆E
j
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Calculation of excited states[7]
Going back to the equation:
ˆ
hφi|(ĤN − ∆)|(φ
0+
X
cj φj )i = ci∆E
j
It is a matrix diagonalization → we can get more than one root:
ˆ α)|(φ0 +
hφi|(ĤN − ∆
X
α
α
cα
j φj )i = ci ∆E
j
ˆ in most cases depends on ci and/or ∆E
Note that ∆
↓
it will be also state dependent
↓
different matrix must be diagonalized for different states
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Calculation of excited states[7]
Equation for AQCC/ACPF:
X
α
α α
cα
i hφj |ĤN + δij (1 − G)∆E |φi i = ∆E cj
i
There is different shifts for all states which depends on the respective
reference energy.
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P.G. Szalay: The AQCC method
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Test calculations on the excited states of diatomic
molecules[8]
Strategy: Compare calculated properties to reliable experimental data
60 electronic states of the following molecules:
• B2, C2, N2, O2
Properties investigated:
• equilibrium geometry (re)
• harmonic vibrational frequency (ωe)
• excitation energy (Te)
• dissociation energy (De)
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Test calculations on the excited states of diatomic
molecules[8]
Errors due to finite basis sets have been eliminated by extrapolating
correlation consistent basis set results.
Basis sets used:
• cc-pVTZ
• cc-pVQZ
• (cc-pV5Z)
• TQ: extrapolated from cc-pVTZ and cc-pVQZ
• (Q5: extrapolated from cc-pVQZ and cc-pV5Z)
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P.G. Szalay: The AQCC method
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Absolute error of excitation energies (TQ basis)
10
MR-AQCC/TQ
MR-CI/TQ
MR-CI+Q/TQ
9
8
Frequency
7
6
5
4
3
2
1
0
-1200 -1000
-800
-600
-400
-200
0
200
400
600
800
1000
1200
-1
Error (cm )
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Absolute error of equilibrium geometry (TQ basis)
16
MR-AQCC/TQ
MR-CI/TQ
MR-CI+Q/TQ
15
14
13
12
11
Frequency
10
9
8
7
6
5
4
3
2
1
0
-0.001
0.0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
Error (Angstrom)
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Absolute error of harmonic vibrational frequency (TQ basis)
14
MR-AQCC/TQ
MR-CI/TQ
MR-CI+Q/TQ
13
12
11
10
Frequency
9
8
7
6
5
4
3
2
1
0
-30
-25
-20
-15
-10
-5
0
5
10
15
20
25
30
-1
Error (cm )
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P.G. Szalay: The AQCC method
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Dissociation energy of O2 and C2 (cm−1 )
Method
MR-AQCC
MR-CISD
MR-CISD+Q
MR-ACPF
exp.
MR-AQCC
MR-CISD
MR-CISD+Q
exp.
cc-pVTZ
40080
39300
40320
40290
cc-pVQZ
O2
41330
40420
41680
41580
48536
48600
48534
C2
49425
49808
49883
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cc-pV5Z
(TQ)
(Q5)
41800
40770
42100
42070
42250
41240
42550
42530
42270
41130
42610
42580
42180
50212
50553
50024
50737
50589
50868
50507
51140
50121
50764±161
22
P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Test calculations on the excited states of diatomic
molecules[8]
Conclusions from these tests:
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Test calculations on the excited states of diatomic
molecules[8]
Conclusions from these tests:
• MR-AQCC is more accurate then MR-CISD,
a smaller reference space is sufficient
• MR-ACPF tends to overestimate higher excitation effects
• Accuracy of MR-AQCC (basis set limit, valence CAS reference):
–
–
–
–
excitation energy: ∼ 300 cm−1
equilibrium bond length: ∼ 0.003 Å
harmonic frequency: ∼ 10 cm−1
dissociation energy: ∼ 200 cm−1
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Test calculations on the excited states of diatomic
molecules[8]
Conclusions from these tests:
• MR-AQCC is more accurate then MR-CISD,
a smaller reference space is sufficient
• MR-ACPF tends to overestimate higher excitation effects
• Accuracy of MR-AQCC (basis set limit, valence CAS reference):
–
–
–
–
excitation energy: ∼ 300 cm−1
equilibrium bond length: ∼ 0.003 Å
harmonic frequency: ∼ 10 cm−1
dissociation energy: ∼ 200 cm−1
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23
P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Test calculations on the excited states of diatomic
molecules[8]
Conclusions from these tests:
• MR-AQCC is more accurate then MR-CISD,
a smaller reference space is sufficient
• MR-ACPF tends to overestimate higher excitation effects
• Accuracy of MR-AQCC (basis set limit, valence CAS reference):
–
–
–
–
excitation energy: ∼ 300 cm−1
equilibrium bond length: ∼ 0.003 Å
harmonic frequency: ∼ 10 cm−1
dissociation energy: ∼ 200 cm−1
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P.G. Szalay: The AQCC method
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Calculation of the transition moment[7]
What else do we need than excitation energies?
The transition probability depends on transition moment:
µα→β ∼ hΨα|µ̂|Ψβ i
with µ̂ being the dipole operator.
For CI the calculation of the transition moment is straightforward.
For AQCC/ACPF (actually all CEPA methods) no wave function exists (we
have modified the equations not the wave function!!!)
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P.G. Szalay: The AQCC method
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Calculation of the transition moment[7]
What else do we need than excitation energies?
The transition probability depends on transition moment:
µα→β ∼ hΨα|µ̂|Ψβ i
with µ̂ being the dipole operator.
For CI the calculation of the transition moment is straightforward.
For AQCC/ACPF (actually all CEPA methods) no wave function exists (we
have modified the equations not the wave function!!!)
Therefore so called Linear Response Theory has to be used
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P.G. Szalay: The AQCC method
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Calculation of the transition moment[7]
Linear Response version of MR-AQCC/MR-ACPF:
X
0
α α
dα
i hφj |ĤN + δij (1 − G)∆E |φi i = ∆E dj
i
Difference to original MR-AQCC/MR-ACPF equations:
• The shift includes the correlation energy of the ground state instead of
the state in question
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P.G. Szalay: The AQCC method
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Calculation of the transition moment[7]
Transition property for variational methods (Christiansen et al. [9]):
X
T0α
=
1 X ∂ 2(E0 + F ) α X 0α
di =
Γt xt.
2 i
∂ci∂εX
t
with Γ0α
t is the transition density for the excitation 0 → α and given by:
Γ0α
t
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=
1 X ∂Γt α
d
2 i ∂ci i
26
P.G. Szalay: The AQCC method
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Calculation of the transition moment[7]
Transition property for variational methods (Christiansen et al. [9]):
X
T0α
=
1 X ∂ 2(E0 + F ) α X 0α
di =
Γt xt.
2 i
∂ci∂εX
t
with Γ0α
t is the transition density for the excitation 0 → α and given by:
Γ0α
t
=
1 X ∂Γt α
d
2 i ∂ci i
For MR-AQCC/MR-ACPF:
Γ0t α = a0
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CI
Γ0t α + (1 − a0)
ref
Γ0t α
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P.G. Szalay: The AQCC method
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Calculation of the transition moment[7]
For MR-AQCC/MR-ACPF:
Γ0t α = a0
CI
Γ0t α + (1 − a0)
ref
Γ0t α
The calculation of transition density requires:
•
ref
•
CI
Γ0t α: transition density in the reference space
Γ0t α: CI-like transition density calculated
MR-AQCC/MR-ACPF coefficients
For more details see:
P.G. Szalay, Th. Müller and H. Lischka, PCCP, 2, 2067 (2002).
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Deviation of total energies for the ground state and of excitation energies with
respect to the full CI results for CH2a)
Method
1 1 A1
2 1 A1
3 1 A1
1 1 B1
2 1 B1
1 1 B2
2 1 B2
1 1 A2
2 1 A2
Full CIb
MCSCF
MR-CISD
MR-AQCC
4.66
6.51
7.70
8.02
1.79
8.91
5.85
10.04
133.1
0.21
-0.30
-0.16
0.43
0.18
-0.20
0.79
0.20
5.2
0.01
-0.03
0.01
0.06
0.01
-0.02
0.07
0.00
2.2
0.00
-0.03
-0.02
-0.02
0.00
-0.03
0.00
-0.03
MR-AQCCLRT
2.2
0.01
-0.04
-0.03
0.00
0.01
-0.03
0.03
-0.04
MR-ACPF
MR-ACPF
LRT
0.1
0.00
-0.04
-0.09
-0.05
0.00
-0.06
0.00
-0.10
0.1
-0.01
-0.03
-0.08
-0.10
0.00
-0.05
-0.06
-0.09
a
The deviation of the ground state energy (E − EF CI ) is given in mHartree, that of the
excitation energy (∆E − ∆EF CI ) is given in eV. The cc-pVDZ augmented basis set has
been used, see text for details.
b
Koch et al. c Amor et al.
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Deviation of excitation energies from the corresponding full CI
results for the CH+a)
Method
Full CIb
6x4 CAS
MR-CISD
MR-AQCC
MR-AQCC-LRT
MR-ACPF
MR-ACPF-LRT
3x2 CAS
MR-CISD
MR-AQCC
MR-AQCC-LRT
MR-ACPF
MR-ACPF-LRT
a
b
21Σ+
8.55
31Σ+
13.52
41Σ+
17.29
11Π
3.23
21 Π
14.13
0.01
0.01
0.01
0.00
0.00
0.02
0.01
0.01
-0.01
-0.01
-0.04
-0.06
-0.06
-0.12
-0.09
0.00
0.00
0.00
0.00
0.00
0.02
0.01
0.01
-0.02
-0.01
0.05
0.00
0.02
-0.14
-0.05
0.05
0.03
0.04
-0.01
0.01
The deviation of the excitation energy (∆E − ∆EF CI ) is given in eV.
Excitation energies are given in eV.
Eötvös Loránd University, Institute of Chemistry
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P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Deviation of transition moments from the corresponding full CI
results for the CH+a)
Method
Full CIb
6x4 CAS
MR-CISD
MR-AQCC
MR-AQCC-LRT
MR-ACPF
MR-ACPF-LRT
3x2 CAS
MR-CISD
MR-AQCC
MR-AQCC-LRT
MR-ACPF
MR-ACPF-LRT
a
b
21Σ+
0.1577
31Σ+
1.0394
41Σ+
0.827
11 Π
0.2992
21 Π
0.7671
-0.0012
-0.0007
-0.0007
0.0003
0.0003
-0.0084
-0.0039
-0.0040
0.0038
0.0042
+0.0002
+0.0002
0.0000
-0.0016
0.0027
-0.0001
-0.0001
-0.0001
-0.0002
-0.0002
-0.0014
-0.0010
-0.0011
-0.0004
-0.0005
0.0040
0.0009
0.0014
-0.0088
-0.0058
0.0128
0.0042
0.0050
-0.0172
-0.0139
The error of the transition moment with respect to full CI in a.u.
Transition moments in a.u.
Eötvös Loránd University, Institute of Chemistry
30
P.G. Szalay: The AQCC method
Eötvös Loránd University, Institute of Chemistry
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
31
P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Deviation of transition moments from the corresponding full CI
1
a
results for the X1Σ+
g → A Πu transition of C2
Method
Full CIb
MR-CISD
MR-AQCC
MR-AQCC-LRT
MR-ACPF
MR-ACPF-LRT
MR-CISD
MR-AQCC
MR-AQCC-LRT
MR-ACPF
MR-ACPF-LRT
Reference
8x8 CAS
4x6 CAS
Distance (Bohr)
2.2
2.6
3.0
0.34402 0.28081 0.14890
-0.00003 0.00053 0.00619
-0.00001 -0.00065 0.00015
-0.00030 -0.00095 0.00002
0.00013 -0.00122 -0.00274
-0.00035 -0.00160 -0.00294
0.03796 0.03929
–c
0.00392 0.00269
–
0.00379 0.00224
–
-0.01986 -0.02539
–
-0.02012 -0.02534
–
a The 3s2p1d ANO basis basis set has been used. The transition moment deviations (T X − T X (F CI)) are given in a.u.
0α
0α
b Transition moments are given in a.u. (from Bauschlicher et al.).
c At 3.0 Bohr the 4x6 CAS reference space is too small to describe the ground state.
Eötvös Loránd University, Institute of Chemistry
32
P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Conclusions on excited state calculations[7]
• MR-AQCC/MR-ACPF is preferred over MR-CISD:
– smaller reference space is sufficient
• MR-AQCC is preferred over MR-ACPF
– MR-ACPF tends to overestimate higher excitation effects
• Accuracy of MR-AQCC
– excitation energy: 0.01-0.03 eV
– transition moment: 1-2%
Eötvös Loránd University, Institute of Chemistry
33
P.G. Szalay: The AQCC method
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
References
[1] P. G. Szalay, Int. J. Quantum Chem. 55, 151 (1995).
[2] P. G. Szalay, Towards state-specific formulation of multireference coupled-cluster
theory: Coupled electron pair approximations (CEPA) leading to multireference
configuration interaction (MR-CI) type equations, in Modern Ideas in Coupled-Cluster
Methods, edited by R. J. Bartlett, pages 81–123, Singapore, 1997, World Scientific.
[3] R. J. Gdanitz and R. Ahlrichs, Chem. Phys. Lett. 143, 413 (1988).
[4] P. G. Szalay and R. J. Bartlett, Chem. Phys. Lett. 214, 481 (1993).
[5] P. Pulay, Int. J. Quantum Chem. S17, 257 (1983).
[6] J. L. Heully and J. P. Malrieu, Chem. Phys. Lett. 199, 545 (1992).
[7] P. G. Szalay, T. Müller, and H. Lischka, Phys. Chem. Chem. Phys. 2, 2067 (2000).
[8] T. Müller, M. Dallos, H. Lischka, Z. Dubrovay, and P. G. Szalay, Theor. Chem. Acc
105, 227 (2001).
[9] O. Christiansen, P. Jørgensen, and C. Hättig, Int. J. Quantum Chem. 68, 1 (1998).
Eötvös Loránd University, Institute of Chemistry
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