Transcript Beyond Hartree-Fock

Post Hartree-Fock Methods in Quantum
Chemistry
Sourav Pal
National Chemical Laboratory
Pune- 411 008
General classification of theoretical chemistry
approaches:
Classical Mechanics (CM)
Quantum Mechanics (QM)
Molecular Mechanics (MM)
Ab initio methods
Semi-empirical
methods
Density functional theory (DFT)
THEORETICAL MODEL
CHEMISTRY
• Should Include Electron Correlation ( twoelectron repulsion effects) in an efficient manner.
• Applicability Should be General
• Results Should Scale Correctly with Number of
Electrons N (Size) e.g. energy proportional to N
• Dissociate into Fragments Correctly
• Accuracy; Computationally Tractable ; Ab initio
description
Electronic Structure Models
• Hartree-Fock Method ( one-particle approximation) sufficient
in many cases; amenable to simple interpretation of molecular
orbital theory. The MO theory overweights the ionic parts and
thus restricted HF fails to describe dissociation closed shell
molecules into open fragments.
• In many cases high level of electron correlation arising from
two-electron repulsion needed calling for post-HF rigorous
developments
• Configuration interaction, perturbation theory and coupledcluster methods are the methods of choice
• Among these, coupled-cluster has emerged as a compact
method to include correlation and describe size-dependence,
dissociation correctly
What is electron correlation and why do we need it?
0 is a single determinantal wavefunction.
 SD 
1 1
1 2
 2 (1)
 2 (2)
 N (1)
 N (2)
1 N   2 (N)
 N (N)
,
 i |  j  ij
Slater Determinant
Recall that the SCF procedure accounts for electron-electron repulsion by
optimizing the one-electron MOs in the presence of an average field of the
other electrons. The result is that electrons in the same spatial MO (antiparallel spins) are too close together. Their motion is actually correlated.
Correlation of anti-parallel spins missing in Hartree-Fock theory
Eel.cor. = Eexact - EHF
(B.O. approx; non-relativistic H)
Electron Correlation
• Instantaneous repulsion between electrons,
missing in mean field or Hartree Fock
method
• Correlation between electrons of opposite
spins, making them avoid each other
• Virtual orbitals in Hartree-Fock method
used for expanding the many-electron wave
function in terms of configurations
(determinants)
• One way to see why simple MO theory does not work
is that at dissociation, more than one determinant is
important
• Any post HF method, based on simple RHF method,
may not work, in general.
• Post-HF expansion must work on multi-determinant
in such cases to correct the problem, in general.
• The state-of-the-art rigorous method is multireference coupled-cluster theory, applicable to high
accuracy results for any state, away from equilibrium
(including dissociation), excited states etc.
• Rigorous method for molecular interaction, properties
and reactivity
Size Consistent and Size Extensive
Size-consistent method - the energy of two molecules (or fragments)
computed at large separation (100 Å) is equal to the twice energy of the
individual molecule (fragment). Only defined if the molecules are noninteracting.
EAB ( R AB) E A + E B
Size-extensive method - the energy scales linearly with the number of
particles.
1.
Full CI is size consistent and extensive.
2.
All forms of truncated CI are not. (Some forms of CI,
esp. MR-CI are approximately size consistent and size
extensive with a large enough reference space.)
RHF dissociation problem
Consider H2 in a minimal basis composed of one atomic 1s orbital on
each atom. Two AOs (c) leads to two MOs ()…
H
H
2  N 2 ( cA  c B );
antibonding MO
H 1s
H 1s
1  N1 ( cA  cB );
H
H
bonding MO
The ground state wavefunction is:
1(1) 1 (1)
0 
1(2) 1 (2)
 0  1 (1)1(2)  1(2)1 (1)
Slater determinant with two electrons
in the bonding MO
Expand the Slater Determinant
 0  1 (1)1 (2) (1)(2)  (1)(2)
Factor the spatial and spin parts
 0  1 (1)1 (2)  ( cA (1)  c B (1))(c A (2)  c B (2)) H does not depend on spin
Four terms in
 0  cA (1) cA (2)  c B (1) cB (2)  c A (1) cB (2)  c B (1) cA (2)
the AO basis
cA c A
cB c B
cA c B
cB c A
Ionic terms, two electrons in one Atomic Orbital
Covalent terms, two electrons shared between two AOs
H2 Potential Energy Surface
.
E
. At the dissociation
H + H
0
H
H
Bond stretching
limit, H2 must separate
into two neutral atoms.
H H
At the RHF level, the wavefunction, , is 50% ionic and 50% covalent at all
cA c B
cA c A
bond lengths.
cB c B
cB c A
H2 does not dissociate correctly at the RHF level!!
Should be 100% covalent at large internuclear separations.
RHF dissociation problem has several consequences:
•
Energies for stretched bonds are too large. Affects transition state structures Ea are overestimated.
•
Equilibrium bond lengths are too short at the RHF level. (Potential well is too
steep.) HF method ‘overbinds’ the molecule.
.
•
•
The wavefunction contains too much ‘ionic’ character; causing dipole
moments (and also atomic charges) at the RHF level to be too large.
However, SCF procedures recover ~99% of the total electronic energy around
equilibrium.
But, even for small molecules such as H2, the remaining fraction of the energy - the
correlation energy - is ~110 kJ/mol, on the order of a chemical bond.
To overcome the RHF dissociation problem,
Use two-configurational trial function that is a combination
of 0 and 1
First, write a new wavefunction using the anti-bonding MO.
2  N 2 ( cA  c B );
antibonding MO
The form is similar to 0, but describes an excited state:
 2(1) 2 (1)
1 
  2(1)2 (2)  2 (2) 2 (1)
 2(2) 2 (2)
1  2 (1) 2 (2)(1) (2)  (1) (2)
MO basis
1  2 (1) 2 (2)  ( cA (1)  c B (1))( c A (2)  cB (2))
1  cA (1) c A (2)  cB (1) c B (2)  c A (1) cB (2)  cB (1) cA (2)
Ionic terms
Covalent terms
AO basis
Configuration interaction
• Linear expansion of wave function in terms
of determinants, classified as different ranks
of hole-particle excited determinants
• Matrix eigen-value equation
• Very accurate determination of a few lowest
eigenvalues using iterative techniques
• Problem of proper scaling with size
Trial function - Linear combination of 0 and 1;
two electron configurations.
  a0  0  a11  a0 (11)  a1(2 2 )
  (a0  a1 )cA c A  c B cB  (a0  a1 )cA c B  c B cA 
Ionic terms
Covalent terms
Three points:
1. As the bond is displaced from equilibrium, the coefficients (a0, a1) vary
until at large separations, a1 = -a0: Ionic terms disappear and the molecule
dissociates correctly into two neutral atoms. The above wave function is
an example of configuration interaction.
2.
The inclusion of anti-bonding character in the wavefunction allows the
electrons to be further apart on average. Electronic motion is correlated.
3.
The electronic energy will be lower (two variational parameters).
Configuration Interaction
Since the HF method yields the best single determinant wavefunction
and provides about 99% of the total electronic energy, it is commonly
used as the reference on which subsequent improvements are based.
As a starting point, consider as a trial function a linear combination of Slater
determinants:
  a0  HF   ai i
Multi-determinant wavefunction
i 1
a0 is usually close to 1 (~0.9).
• M basis functions yield M molecular orbitals.
• For N electrons, N/2 orbitals are occupied in the RHF wavefunction.
• M-N/2 are unoccupied or virtual (anti-bonding) orbitals.
Generate excited Slater determinants by promoting up to N
electrons from the N/2 occupied to M-N/2 virtuals:
b
a
9
a,b,c… =
virtual MOs
a
8
a,b
b
a
b
a
c
c,d
k
i
k,l
i
j
j
abc
ijk
abcd
ijkl
7
6
5
i
i
4
i,j
3
i,j,k… =
occupied MOs 2
j
1
HF
Excitation level
Ref.
ia
Single
ijab
Double
ijab
Triple
Quadruple
…
Represent the space containing all N-fold excitations by (N).
Then the COMPLETE CI wavefunction has the form
CI  C0 HF  (1)  (2)  (3)  ... (N )
Where
 HF  Hartree Fock
oc c vi rt
    Cia ia
i a
(1)

(2)
oc c vi rt
   Cijab ijab
i, j

(3)
a,b
oc c vi rt

C
i, j ,k a,b,c

(N )
Linear combination of Slater
determinants with single excitations
Doubly excitations

oc c
abc
ij k
vi rt
 
i, j ,k.. . a,b,c...
ijabc
k
abc.. .
Triples
abc.. .
Cij k.. . ij k.. .
N-fold excitation
The complete CI expanded in an infinite basis yields the exact solution
to the Schrödinger eqn. (Non-relativistic, Born-Oppenheimer approx.),
often used as benchmark.
abc...
The various coefficients, Cijk... , may be obtained in a variety of ways.
A straightforward method is to use the Variation Principle.
ECI 
E CI



CI | H | CI
CI | CI
C
abc...
ijk...
0
HCK  E K CK
Expectation value of He.
Energy is minimized
wrt coeff
In a fashion analogous to the HF eqns,
the CI Schrodinger equation can be
formulated as a matrix eigenvalue
problem.
abc...
The elements of the vector, CK , are the coefficients, Cijk...
And the eigenvalue, EK, approximates the energy of the Kth state.

E1 = ECI for the lowest state of a given symmetry and spin.
 and spin, and so on.
E2 = 1st excited
state of the same symmetry
Configuration State Functions
Consider a single excitation from the RHF
reference.
Both RHF and (1) have Sz=0,
but (1) is not an eigenfunction
of S2.
RHF
(1)
Linear combination of singly excited
determinants is an eigenfunction of S2.
Configuration State Function, CSF
(Spin Adapted Configuration, SAC)
Singlet CSF
Only CSFs that have the same multiplicity
as the HF reference contribute to the
correlation energy.
1,2  1(1) 2(2)  1 (2) 2 (1)
Multi-configuration Self-consistent Field (MCSCF)
9
Carry out Full CI and orbital optimization within a
small active space. Six-electron in six-orbital MCSCF
is shown. Written as [6,6]CASSCF.
8
7
Complete Active Space Self-consistent Field (CASSCF)
6
H2O MOs
5
4
3
2
1
HF
Why?
1. To have a better description of the ground or
excited state. Some molecules are not welldescribed by a single Slater determinant, e.g. O3.
2. To describe bond breaking/formation; Transition
States.
3. Open-shell system, especially low-spin.
4. Low lying energy level(s); mixing with the
ground state produces a better description of the
electronic state.
5. …
MCSCF Features:
1.
In general, the goal is to provide a better description of the main features of
the electronic structure before attempting to recover most of the correlation
energy.
2.
Some correlation energy (static correlation energy) is recovered. (So called
dynamic correlation energy is obtained through CI and other methods
through a large N-particle basis.)
3.
The choice of active space - occupied and virtual orbitals - is not always
obvious. (Chemical intuition and experience help.) Convergence may be
poor.
4.
CASSCF wavefunctions serve as excellent reference state(s) to recover a
larger fraction of the dynamical correlation energy. A CISD calculation
from a [n,m]-CASSCF reference is termed Multi-Reference CISD (MRCISD). With a suitable active space, MRCISD approaches Full CI in
accuracy for a given basis even though it is not size-extensive or consistent.
Examples of compounds that require MCSCF
for a qualitatively correct description.
H
H
C
C
O+
H
H
O
O
O-
O
zwitterionic
Singlet state of twisted
ethene, biradical.
biradical
H
C
N
H
C
N
Transition State
O
H
C
N
Mœller-Plesset Perturbation Theory
In perturbation theory, the solution to one problem is expressed in terms of
another one solved previously. The perturbation should be small in some
sense relative to the known problem.
Hˆ  Hˆ 0  Hˆ '
Hˆ 0  i  E i i, i = 0,1,2,...,
Hamiltonian with pert., 
Unperturbed Hamiltonian
Hˆ   W
W   0W0  1W1   2W2  ...
   00  11   22  ...
As the perturbation is turned on, W (the
energy) and  change. Use a Taylor series
expansion in .
ˆ and Hˆ '
Define H
0
N


ˆ
ˆ
ˆ
ˆ
ˆ
H0   Fi   hi   J ij  K ij 

i 1
i1 
j 1
N
N
N
N

N

N
Hˆ '   gij    gij
i 1 j 1
i 1 j 1
W0  sum over M O energies
W1 =  0| | Hˆ '|  0  E(HF)
oc c vi r
W2   
ab
ab
 0| | Hˆ '|  ij  ij | Hˆ '|  0
E 0  E ijab
i  j a b


E(MP2)   
oc c vi r
i j a b
i
j
Unperturbed H is the sum over Fock
operators  Moller-Plesset (MP) pert th.
Perturbation is a two-electron
operator when H0 is the Fock
operator.
With the choice of H0, the first
contribution to the correlation
energy comes from double
excitations.
| a  b   i j |  b a
i   j  a  b

2
Explicit formula for 2nd
order Moller-Plesset
perturbation theory, MP2.
Advantages of MP’n’ Pert. Th.
• MP2 computations on moderate sized systems (~150 basis functions)
require the same effort as HF. Scales as M5, but in practice much less.
• Size-extensive (but not variational). Size-extensivity is important; there is
no error bound for energy differences. In other words, the error remains
relatively constant for different systems.
• Recovers ~80-90% of the correlation energy.
• Can be extended to 4th order: MP4(SDQ) and MP4(SDTQ). MP4(SDTQ)
recovers ~95-98% of the correlation energy, but scales as M7.
• Because the computational effort is significanly less than CISD and the
size-extensivity, MP2 is a good method for including electron correlation.
Coupled Cluster Theory
Perturbation methods add all types of corrections, e.g., S,D,T,Q,..to a given
order (2nd, 3rd, 4th,…).
Coupled cluster (CC) methods include all corrections of a given type to infinite
order.
The CC wavefunction takes on a different form:
ˆ
T
CC  e 0
Coupled Cluster Wavefunction
0 is the HF solution
1 ˆ 2 1 ˆ3
ˆ
ˆ
e 1T + T  T 
2
6
Tˆ
ˆ T
ˆ  Tˆ  Tˆ 
T
1
2
3

1ˆk
 T
k 0 k!
 Tˆ N
Exponential operator generates
excited Slater determinants
Cluster Operator
N is the number of electrons
CC Theory cont.
oc c vi r
ˆ     t a a
T
1 0
i
i
i
a
oc c vi r
ab ab
ˆ  
T
t


2 0
ij  ij
The T-operator acting on the HF reference
generates all ith excited Slater Determinants, e.g.
doubles ijab.
i j a b
tia
ab
ij
t

Expansion coefficients are called amplitudes; equivalent to
the ai’s in the general multi-determinant wavefunction.
HF ref.
ˆ
1 ˆ 2  ˆ
ˆ
ˆ T
ˆ  1 Tˆ 3   T
ˆ  Tˆ T
ˆ  1 Tˆ 2  1 Tˆ T
ˆ2 1 T
ˆ 4  
e T  1ˆ  Tˆ1  T

T

T

T
2
2 1
3 1
2 1   3
6 1   4
2 2 2 2 1 24 1 
singles
doubles
triples
Quadruple excitations
The way that Slater determinants are generated is rather different…
CC Theory cont.
1ˆ
Tˆ
1
1 ˆ 2 
ˆ
T

 2 2 T1 
ˆ
ˆ Tˆ  1 T
ˆ 3 
T

T
2 1
 3
6 1 
ˆ
T
2
ˆ2
T
1
ˆ
T
3
ˆ Tˆ
T
2 1
ˆ3
T
1
ˆ
ˆ  1 Tˆ 2 
T4  Tˆ 3 T
1
2


2


1
1
2
4
ˆ 
ˆ 
 Tˆ T
T
2
1
 2
24 1 
HF reference
Singly excited states
Connected doubles
Dis-connected doubles
Connected triples, ‘true’ triples
‘Product’ Triples, disconnected triples
ˆ
T
4
True quadruples - four electrons interacting
ˆ 2
T
2
Product quadruples - two noninteracting pairs
ˆ Tˆ , T
ˆ Tˆ 2, Tˆ 4
T
3 1 2 1
1
Product quadruples, and so on.
CC Theory cont.
If all cluster operators up to TN are included, the method yields energies that are
essentially equivalent to Full CI.
In practice, only the singles and doubles excitation operators are used forming
the Coupled Cluster Singles and Doubles model (CCSD).
ˆ
ˆ
ˆ  T
ˆ 1T
ˆ 2  Tˆ T
ˆ  1 Tˆ 3   1 Tˆ 2  1 Tˆ T
ˆ2 1 T
ˆ 4 
e T1  T2  1ˆ  T
1
 2 2 1   2 1 6 1  2 2 2 2 1 24 1 
The result is that triple and quadruple excitations also enter into the energy
expression (not shown) via products of single and double amplitudes.
It has been shown that the connected triples term, T3, can be important. It can be
included perturbatively at a modest cost to yield the CCSD(T) model. With the
inclusion of connected triples, the CCSD(T) model yields energies close to the
Full CI in the given basis, a very accurate wavefunction.
Comparison of Models
Scale with M
Size-extensive/consist ent
Variat ional
Generally applicable
Requires ŌgoodÕzero-order 
Extension to Mult i-reference
CI-SD
M6
No
Y
Y
Y
Yes
CI-SDT Q
M10
~Yes
Y
No
~No
MP2
M5
Y
No
Y
Y
Yes
MP4(SDT Q)
M7
Y
No
Y
Y
CCSD
M6
Y
No
Y
~No
Not yet
common
CCSD(T)
M7
Y
No
Y
No
Accuracy with a medium sized basis set (single determinant reference):
HF << MP2 < CISD < MP4(SDQ) ~CCSD < MP4(SDTQ) < CCSD(T)
In cases where there is (a) strong multi-reference character and (b) for excited
states, MR-CI methods may be the best option.
Size-dependence
• Dimer of non-interacting H2 molecules
• D-CI wave function contains doubly excited
determinants on each H2 molecule , but not the
quadruply excited determinant
• The product of monomer D CI, however, contains
the quadruply excited
• Φab
Φcd
x
• However, exponential wave function of dimer
even at doubles level contains quadruply excited
determinant
MOLECULAR PROPERTIES
• Defined as derivatives of molecular energy with respect to
perturbation parameters.
EXAMPLES: DIPOLE MOMENT, POLARIZIBILITY,
NUCLEAR GRADIENTS AND HESSIANS ETC.,
• Finite-field (numerical) method.
– Numerical evaluation of energy derivatives by
computing energy at least two different fields.
– Evaluate (no extra effort other than calculating energy)
– Very inaccurate as it involves taking difference of two
large. numbers.
• ANALYTIC METHOD
– Closed analytic form for energy derivatives.
– Requires extra effort to solve for energy
– Response equations.
– easier evaluation of energy and property surfaces.
• Properties for variational (stationary) theories
– S.Pal, TCA 66, 151(1984); PRA 34, 2682(1986); PRA 33,
2240(1986); TCA 68, 379(1985); Vaval, Ghose and Pal, JCP 101,
4914 (1994); Vaval & Pal, PRA 54, 250(1996).
Coupled-Cluster Response
• H()=H+  O
• Equations for different order response:
• E (1) = <  | exp(-T) {O +[H,T (1) ]} exp (T)|  >
• 0 = <  * | exp(-T) {O+[H,T (1) ]} exp (T)|  >
• Linear Equation to be solved for T (1) amplitudes
• Due to multi-commutator expansion, extensivity of
properties retained
• Properties with coupled cluster method
– Non-variational theory
– No Hellmann-Feynman theorem
– Energy first derivative depends on first
derivatives of cluster amplitudes which
have to be explicitly computed.
E(1) = YT T(1) + …. y T,A (Perturbation
independent)
AT(1) = B
B Perturbation independent
Z-VECTOR
METHOD FOR SRCC
–Recasting energy derivative expression to eliminate perturbation
dependent cluster derivative in favour of perturbation independent zvector
E(1) = yTA-1 B = ZT B ( ZT set of de-excitation amplitudes)+..
Z-vector needs
to be solved only once.
ZTA = yT (Perturbation independent)
Dalgarno and Stewart, Handy and Schaefer
Z-Vector can be introduced by stationary approach using Lagrange
multipliers
£ = E -  ZiT < i* | exp(-T) H exp (T)|  >
£ /  T = 0 provides Z amplitudes same as non-var
Case of near-degeneracy
• At away from equilibrium, more than one configuration is
important (non-dynamic correlation)
• Perturbative or cluster expansion around any one
determinant causes convergence problem
• Coupled-cluster equations based on single reference suffer
from convergence (intruder)
• A correct way to solve the problem is to start from the a
space of important determinants and the exponential waveoperator to generate dynamic correlation
• Several different versions of multi-reference CC theory
• Hilbert and Fock space, State-selective type
Multi-reference coupled-cluster
• For near-degenerate systems, methods with starting space
consisting of a linear combination of multi-determinants and
an exponential operator acting on this space constitute a class
of multi-reference coupled-cluster theories
• Very powerful method with proper combination of dynamic
and non-dynamic electron correlation
• The nature of T operator can vary in this case, depending on
the nature of the MRCC method.
• Multiple roots obtained by diagonalisation of an effective
Hamiltonian over the starting model space
• Contribution to the Fock space MRCC, ideal for ionization/
excitation energies etc.
•
(Mukherjee D and Pal S, Adv Quant Chem 20, 291 (1989); Pal et al, J.
Chem. Phys. 88, 4357 (1988); Vaval and Pal, J. Chem. Phys. 111, 4051
(1999)
MULTI-REFERENCE THEORIES :
EFFECTIVE HAMILTONIAN
APPROACH
• Define quasi-degenerate model space p
• P° =  l> <  |
|(0)  =  Ci | 
• Transform Hamiltonian by ‘’ to obtain an effective
Hamiltonian such that it has same eigen values as the
real Hamiltonian.
P° Heff P° = P°H  P°
(Heff)ij Cj = ECi
• Obtain energies of all interacting states in model
space by diagonalizing effective Hamiltonian over
small dimensional model space p
MULTI-REFERENCE THEORIES :
EFFECTIVE HAMILTONIAN
APPROACH
• Bloch equation
H  =  Heff
• Coupled cluster anastz for wave operator 
 = exp(T)
• P[H  -  Heff ] P = 0
Q[H  -  Heff ] P = 0
Heff C = C E
• Multiple states at a time at a particular geometry
Variants of Multi-reference CC
• Effective Hamiltonian theory: Effective Hamiltonian over the
model space of principal determinants constructed and energies
obtained as eigen values of the effective Hamiltonian
• Valence-universal or Fock space: Suitable for difference energies
( Mukherjee, Kutzelnigg, Lindgren, Kaldor and others)
Common vacuum concept; Wave-operator consists of holeparticle creation, but also destruction of active holes and particles
contained in the model space
•
State-universal or Hilbert space: Suitable for the potential energy
surface. Each determinant acts as a vacuum (Jeziorski and
Monkhorst; Jeziorski and Paldus ; Balkova and Bartlett)
Multi- reference coupled cluster thus is more general and powerful
electronic structure theory
To make the theory applicable to energy derivatives like
properties or gradients, Hessians etc., it is important to develop
linear response to the MRCC theory
Eqn for Cluster amplitude derivative and Heff
 =  / { P[H  -  Heff ] P} = 0
 =  / { Q[H  -  Heff ] P} = 0
Eqn. For energy derivative and model space coefficient derivative
Heff (1) C + Heff C(1) = C (1) E + C E(1)
S. Pal, Phys. Rev A 39, 39, (1989); S. Pal, Int. J. Quantum Chem,
41, 443 (1992)
Fock Space Multi-reference Coupled-Cluster
Approach
• ( Mukherjee and Pal, Adv. Quant. Chem. 20, 291 ,1989)
• N-electron RHF chosen as a vacuum, with
respect to which holes and particles are
defined.
• Subdivision of holes and particles into
active and inactive space, depending on
model space
• General model space with m-particles and
n-holes
 (0) (m,n) =  iC i i (m,n)
Fock space MRCC
• P(k,1)[H  -  Heff ] P(k,1) = 0
Q(k,1)[H  -  Heff ] P(k,1) = 0
k = 0, m; 1=0, n
k = 0, m; 1=0, n
Heff is the effective Hamiltonian defined over the model
space determinants. The eigen values of it gives the exact
energies of interest.
• For low-lying excited states one hole one particle model
space is suitable.
Fock Space MRCC
• For the general one active hole and one active particle problem
the model space is written as
|µ(0)(k,1)> = iCi |i(k,1)>
k=0,1; l=0,1
The wave operator will be {exp ( S(k,l)) }
• S (k,1) = Ť(0,0) + Ť(0,1) + Ť(1,0) + Ť(1,1)
• For the (1,1) problem, the model space is an incomplete model
space (IMS). Though generally for IMS, to have linked cluster
theorem, intermediate normalization has to be abandoned, for
(1,1) model space, the equations can be derived assuming
intermediate normalization.
P(1,1)  P(1,1)  P(1,1) + P(1,1) T1 (1,1) T1 (0,0) P(1,1) +……
In addition, for (1,1) model space T1(1,1) operator is
in the wave operator, but it does not contribute to the
energy
and thus can be neglected in the energy derivative or linear
response problem
Computationally full singles and doubles approximation has
been used. For excitation energies closed part of the
connected (H exp(T(0,0)) is dropped, to facilitate direct
evaluation.
Finally to get the singlet and triplet excited states one
diagonalizes the spin integrated effective Hamiltonian
matrices HSEE and HTEE
Z- Vector method for MRCC theory
• In a compact form the response equation may be written as,
• A T (1) = B
• A : Perturbation -independent matrix
•
B : Perturbation-dependent column vector
Two options: Eliminate T(1) in Heff (1)
• Eliminate perturbation-dependent T(1) in energy expression
Elimination in Heff (1) ensures that all roots can be obtained
without using perturbation-dependent vectors, but this
requires a larger no of Z vectors ( square of model space)
D. Ajitha, N. Vaval and S. Pal, J Chem Phys 110, 8236 (1999); J.
Chem. Phys 114, 3380 (2001);
Z-vector solved from a perturbation independent linear equation
Use elimination in energy expression for a single root,
E I (1) =  C' i [Heff (1) ]  Ci
Simplified expression
E I (1) = Y (I) * T (1) + X(I) * F(1) + Q(I)* V(1)
Define Z-vector Z(I)through Matrix equation
Y (I) = Z (I) A
E I (1) expressed in terms of z-vector
E I (1) = Z ( I) * B + X (I) F (1) + Q (I) * V (1)
Z - vector although perturbation independent, still depends
on state of interest
No single Z- vector for all roots at the same time
K R Shamasundar and S. Pal, J. Chem. Phys. 114, 1981 (2001); Int. J. Mol. Sci.
3, 710 (2002)
Analytic linear response
• Analytic linear response for FSMRCC
H(g) = H + gH(1)
Θ = {T, Heff, E, C, Č}  perturbation dependent
Θ(g) = Θ(0) + g Θ(1) + ½!g2 Θ (2) + …..
Θ(n) :: nth order response
Hierarchical equations for response quantities Θ(n) can be
derived.
Specific expressions derived for [0,1], [1,0] and [1,1]
sectors and first order response of energy calculated.
C,Č obey (2n+1) – rule, but the same advantage not
enjoyed by the cluster amplitudes.
Z-vector response approach to FS MRCC
• Early attempt:- Use of a de-excitation vector Z of same
size as total T amplitudes T[0,0] and T[0,1].
Factorization of response equation possible only for T[0,1](1)
and is only for the highest valence case.
For HSMRCC M2 linearly independent de-excitation
amplitudes can eliminate totally all elements of T(1).
Elimination of T(1) can be carried out separately for each
element E(1)
Dependent on C, Č State specific
•
Shamasundar and Pal ; JCP 114, 1981 (2001)
Constrained Variation
• In SRCC, the Z-vector can be introduced by making
energy stationary with non-variational CC equations
as constrains ( Jorgensen et al)
• Along the lines of SRCC, same can be effected by
using variation of ČHeff C with constraints on the
equations for T( Lagrange multiplier).
• The approach used for Hilbert space MRCC
and Fock space MRCC
• Shamasundar and Pal, Int. J Mol. Sci. 4, (2003)
• Shamasundar, Asokan and Pal J. Chem. Phys. 120, 6381 (2004)
Structure of FSMRCC response equations
J (Q )   C [An]C[ nA] H eff
 [ n ]

n
 M
[i ]
i 0
(T [i ] , H eff[i ] , [i ] )
 E A ( C A[ n]C[ nA]  1)

M
[i ]
(T
[i ]
,H ,
[i ]
eff
[i ]
)    
[i ]




|  | c  c | H   H eff
[i ]
 ,l
[i]
l
[i]
l
    |  |   | H   H eff
[i ]
 ,
[i ]
[i ]
[i ]
[i ]
| [i ] 
[i ]
| [i ] 
Structure of FSMRCC response equations
• The stationary equations are obtained by making the
Lagrange functional stationary with respect to the T
amplitudes,  amplitudes and effective Hamiltonian
elements.
[k ]

M
 in C A[ n]C[ nA]  
0

k  i  H eff
n
[i]
 M [k ]
0

[i ]
k i  T
n
Structure of FSMRCC response equations
 C C
n
i
[n]
A
M


k  i  H eff
n
[n]
A
[k ]
[i]
0
Structure of FSMRCC response equations
• M (k) depends on lower-valence T’s. Hence in the
stationary equation with respect to T[i] summation index is
from i to n , where n is the highest valence sector.
• To solve Lagrange multipliers for a specific sector, all
higher valence ’s are necessary.
• SEC decoupling, reverse to the T equations, is present in
the  equations
• In the [i] determining equation, all higher valence  are
present and are in the inhomogeneous part of the equation
• For the equation determining highest valence , the
inhomogeneous part contains model space coefficients and
this makes the theory state-specific
Structure of FSMRCC response equations
• For the highest valence sector, the non-zero
inhomogeneous parts are present only for the closed parts
of  (n)
• Open and closed parts of  are coupled in each FS sector
• Closed part of Lagrange multipliers takes care of
incompleteness of model space
• For incomplete model space, the closed parts appear
explicitly because effective Hamiltonian can not be
defined explicitly in terms of the cluster amplitudes
• Similarity transformation approach can eliminate the
closed parts, but this is not available in general
Structure of FSMRCC response equations
• Special case of incomplete model space/ complete model
space (CMS) results in simplifications. For CMS,
intermediate normalization makes the definition of the
effective Hamiltonian explicit, thus allowing definition of
closed parts of  in terms of open amplitudes of , T and
model space coefficients
• P H P = P Heff P = P Heff P
• This the solution of  amplitudes involves only the open
parts of them
• Similar simplifications appear for quasi-complete model
space e.g. (1,1) Fock space
• For (1,1) model space, the one-body part of (1,1) T and 
amplitudes are not specifically required for energy
derivatives, since connected diagrams are not possible
using these operators.
Rigorous spectra and properties
• Accurate calculation of difference energies using
multi-reference coupled-cluster method (accuracy
within 0.1mev)
• Developed analytic approach based on variational
coupled cluster method for molecular properties of
closed and open shell molecules, excited states.
S. Pal, M.Rittby , R.J.Bartlett, D.Sinha and D.Mukherjee
J.Chem.Phys.,88,4357 (1988); N. Vaval, K.B.Ghose, S. Pal and
D.Mukherjee, Chem.Phys.Lett.,209, 292 (1993); N. Vaval and S. Pal, J.
Chem. Phys 111, 4051(1999); S.Pal, Theor. Chim. Acta., 68, 379 (1985);
Phys.Rev.A, 33, 2240 (1986); Phys. Rev. A. 34, 2682 (1986);
Phys.Rev.A,39,39,(1989); N. Vaval, K.B.Ghose and S. Pal, J. Chem. Phys,
101, 4914, (1994); N. Vaval & S.Pal, Phys.Rev.A 54, 250 (1996); D. Ajitha,
N.Vaval and S. Pal, J. Chem. Phys. 110, 2316 (1999); P. Manohar and
S.Pal, Chem.Phys.Lett. (2007) (In Press)
TABLE I. Adiabatic excitation energies and dipole moments of H2O
using the FSMRCC response approach
Excitati
on
energy
(eV)
Expt.
Excitatio
n energy
(eV)
Total energy at
the FSMRCC in
a.u.
MRCC
FF (a.u.)
MRCC
Anal (a.u.)
CASSC
Pa (a.u.)
1
1( b1
7.287
7.49b
-76.0221
(0.005 a.u.)
-76.0247
(0.00 a.u.)
-76.0284
(-0.005 a.u.)
-0.636
-0.603
-0.712
1
1( b1
6.878
7.0,c7.2d -76.0374
(0.005 a.u.)
-76.0397
(0.00 a.u.)
-76.0426
(-0.005 a.u.)
-0.520
-0.599
-0.478
State
1B
3sa1)
3B
3sa1)
Table II: Dipole moment values of the HCOO
radical using analytic Fock space multi-reference
coupled cluster response approach
Manohar, Vaval and Pal , Theo. Chem., 768, 91 (2006)
Table III: Dipole moment of OH radical (in au)
using FSMRCC method
Manohar, Vaval and Pal , Theo. Chem., 768, 91 (2006)
Table IV: Dipole moment of Nitrogen oxides (in
au) using FSMRCC method
Higher-order energy Derivatives
• £ (g, ) = £ (0) + g £ (1) + 1/ 2 gg £ (2) +…..
• £ (n) is a functional of quantities upto  (m) (m=0,1,2 ..n).
• Response of the quantities  are obtained by making  (m)
stationary with respect to (0) i.e.
 £ (n) /  =0
For first-order response,  £ (1) /  =0, yielding response
quantities.
Use (2n+1) type rule to explicitly compute higher energy
derivatives upto third-order
Structure of first-order response wavefunction
 Sector wise solution of the (1) amplitudes
 Similar structure of Fock space equations for first
derivatives of T and  hold good. (SEC
decoupling starting from lowest sector for T(1)
amplitudes and reverse SEC decoupling for  (1)
amplitudes
 Use (2n+1) type rule to explicitly compute higher
energy derivatives upto third-order
 Expressions derived upto (1,1) sector.
Table V Polarizabilities of OH, HCOO and OOH
radicals
Manohar and Pal, communicated
Thank You
For
Your ATTENTION