1

Electronic transition dipole moments and dipole oscillator strengths within

Fock-space multi-reference coupled cluster framework – An efficient and

novel approach

Debarati Bhattacharya, Nayana Vaval and Sourav Pal *

Physical Chemistry Division,

CSIR - National Chemical Laboratory,

Pune - 411008, India

Abstract:

Within the Fock-space multi-reference coupled cluster framework, we have evaluated the

electronic transition dipole moments, which determine absorption intensities. These depend on

matrix elements between two different wave functions (e.g. ground state to the excited state). We

present two different ways, to calculate these transition moments. In the first method, we

construct the ground and excited state wave functions with the normal exponential ansatz of

Fock-space coupled cluster method and then calculate the relevant off-diagonal matrix elements.

In the second approach, we linearize the exponential form of the wave operator which will

generate the left vector, by use of Lagrangian formulation. The right vector is obtained from the

exponential ansatz. In order to relate the transition moments to oscillator strengths, excitation

energies need to be evaluated. The excitation energies are obtained from the Fock-space multi-

reference framework. The transition dipole moments of the ground to a few excited states,

together with the oscillator strengths of a few molecules, are presented.

Keywords: Fock-space, transition moment, oscillator strength.

Electronic mail : s.pal@ncl.res.in

2

I. INTRODUCTION

Transition dipole moments (TDM) are of great general interest as they determine the

transition rates and probability of photon or electric field induced atomic and molecular state

changes. It is a good test for assessing the validity and accuracy of ab-initio calculations. A

calculation of transition dipole moment can be helpful in understanding the energy transfer rates;

provide a basis for calculating extinction coefficients and fluorescence lifetimes etc.1,2

The

electronic transition dipole moment (ETDM) is an important prerequisite for understanding

optical spectra. The probabilities per unit time for absorption induced emissions and spontaneous

emissions (as derived from the first order, time-dependent perturbation theory in the dipole

length approximation) are proportional to the square of the TDM between the two chosen states

of interest.3 For any transition from a state ‘p’ to state ‘q’, the TDM in the dipole length form is

expressed as –

pq p qd (1)

In order to understand and characterize radiative processes, we have to relate it to experimental

observables (such as oscillator strength). The oscillator strength in the dipole length

approximation is given by, 4

22

3pq pqf E d

where

q pE E E (2)

Given the importance of transition moments, computing them and relating them to experimental

observables is not very straightforward. This is due to the sensitivity of pqd towards the quality

of the wave function.5

As stated previously, the calculation of TDM represents a different test

altogether for any ab-initio method as there can be considerable redistribution of charge in a

molecular situation without substantial change in the energy. Hence, calculation of TDM

demands an accurate description of the wave function.

In the last few decades, coupled cluster (CC) theory6 has proven to be quite successful in

describing ground-state electronic structure of molecules. Compared to other ab-initio methods,

single-reference CC theory can accurately treat the dynamical correlation of electrons in

3

molecular systems. Excitation energies can be calculated using either single-reference (SR)7-13

or

multi-reference (MR) CC14-23

techniques. Within the class of SR methods, excitation energies

can be computed using linear response,7-10

equation-of-motion (EOM) methods11,12

or the

symmetry adapted cluster configuration interaction (SACCI)13

formalism. Among the class of

MRCC methods, excitation energies can be obtained from the Fock-space effective

Hamiltonian14-21

based approach.

The MRCC methods are subdivided into a multi-root effective Hamiltonian24-26

approach

and a state-specific27-29

approach. This class of multi-root effective Hamiltonian method is

further subdivided into a Hilbert space (HS) CC22-23

method and a valence-universal one, also

known as Fock-space (FS) CC14-21

method. The HSCC method is better suited for potential

energy calculations while FSMRCC variant has been proven suitable in cases; where ionization,

electron attachment and electron excitation in molecules occur. Thus, in order to calculate

properties related to excited states, FSMRCC is the preferred method of choice.

Calculation of excitation energies and electronic transition moments in multi-

configuration linear response (MCLR) was done by Olsen et al.30

Within the EOMCC

formalism, Stanton and Bartlett31

presented an idea to calculate the transition probabilities via a

systematic bi-orthogonal approach. Size intensive transition moments from the coupled cluster

singles and doubles linear response (CCSDLR) function was formulated by Jørgensen and co-

workers.32

A detailed description to calculate expectation values and transition elements by

coupled cluster theory in general, was presented by Prasad.33

Integral-direct frequency dependent

polarizabilities and transition probabilities in the CC framework were implemented by

Christiansen et al.34

In the CC2 model, transition moments were computed using the resolution-

of-the-identity approximation by Hättig and Köhn.35

Later on Köhn and Pabst36

implemented

transition moments between excited states using the RI-CC2 approximation. In an early work by

Stolarczyk and Monkhorst,37

derivation of expectation value and transition moment was

formulated within the generalized CC framework. The main idea was to define a new operator

W and calculate the matrix elements of that operator. For an arbitrary n-particle operator- V , the

transition moment was expressed as ˆY X Y

XV V for X Y . In order to calculate the

transition moment, one had to know the operator- 1ˆ ˆ ˆ ˆW V .

4

Barysz et al38,39

implemented the above mentioned scheme of Stolarczyk and Monkhorst

in the FSCC framework to obtain the electronic transition moments and oscillator strength of

certain molecules. They used the CCSD approximation and truncated the W operator at the

quadratic level. In the present paper, we compute the electronic transition moment of the dipole

operator between the ground and a few excited states of some molecules in two different ways.

We start by the initial formula of transition moment where the n-particle operator- V , is chosen

as the one-particle dipole moment operator. Instead of redefining another operator- W , we

calculate the expectation value of the dipole moment operator between the ground and the

excited state wave functions. Hence, only T(0,0)

amplitudes are sufficient to describe the ground

state. The excited state is generated from the Fock-space. In the other method, we take recourse

to the constrained variation approach, where the matrix element is computed using a

biorthogonal approach. This involves solving an extra set of de-excitation -amplitudes to

describe the ground state and all other higher sector -amplitudes to describe the excited state.

The details of the theory are given later. The entire implementation is done in the CCSD

approximation.

It is our purpose here to treat the electronic transition dipole moments, dipole strengths

and oscillator strengths for a few allowed transitions amongst the various excited states of CH+,

BH, H2O and H2CO. The details of the theory are stated in section II. Implementation and

computational aspects follow in section III. Results and discussions are given in section IV.

Concluding remarks are noted in the final section V.

II. THEORY

A. Excitation energy in FSMRCC

The Fock-space method in the CC framework is well described and accounted for

evaluation of excitation energies.17,18

A brief review is presented here followed by the

description of the transition moments within the same.

The basic assumption in the FSCC method is that of a common vacuum. The vacuum is

chosen to be the restricted Hartree-Fock solution of the N-electron state. Particles and holes are

5

defined with respect to this vacuum. There is a further sub-division of the particles and holes into

active and inactive particles and holes. The various sectors in FS are a representation of this

active space. In general the number of active holes and particles are represented by the

superscript of the wave function. A general model space consisting of ‘p’ active particles and ‘h’

active holes is given by –

( , ) ( , )( , )(0)p h p hp h

ii

i

C (3)

where, ( , )p hiC are the combination or model space coefficients of i . The correlated wave

function for a particular th state is given by,

( , ) ( , )(0)

p h p h (4)

In the above equation - is the universal wave operator. The universal wave operator will be

generating states by its action on the reference wave function. has the specific form

( , )T p he

(5)

where the parenthesis denote normal ordering. The ‘T’ is known as the cluster operator and is

expressed as -

( , ) ( , )

0 0

p hp h k l

k l

T T

(6)

where ( , )k lT can create particles and holes, in addition to destroying exactly k active particles and l

active holes. To calculate excitation energies, we define our system as a specific problem of the

(1,1) sector, i.e. one active particle and one active hole. The Schrodinger equation for quasi-

degenerate states in the Fock-space formalism is given by,

(1,1) (1,1)H E (7)

On substituting the above equation with equations 3 and 4 gives,

6

(1,1) (1,1) (1,1)(1,1)i i i i

i i

H E CC

(8)

The states generated by the action of the universal wave operator on the reference space are such

that they satisfy the Bloch equations. An effective Hamiltonian is defined through the Bloch

equations, which is given by

(1,1) (1,1) (1,1)

(1,1) (1,1) (1,1)

0

0

eff

eff

P H H P

Q H H P

(9)

where (1,1)P is the Projection operator for the model space, defined as :

(1,1) (1,1) (1,1)

i ii

P (10)

The complimentary space (1,1)Q is defined in the following manner

(1,1) (1,1)1Q P (11)

Solving the above Bloch equations give us the effective Hamiltonian and the cluster amplitudes.

Normal ordering ensures that the higher sector amplitudes do not occur while solving for the

lower sector ones. While solving for the (1,1) sector the (0,0), (0,1) and (1,0) sector amplitudes

appear as constant entities. This is known as sub-system embedding condition (SEC). There is a

further decoupling in the effective Hamiltonian effH and (1,1)T amplitudes as shown in

references 40 and 41. Spin adaptation separate out the singlet and triplet effective Hamiltonian.

The triplet energy can be obtained by solving only the triplet effective Hamiltonian. But in order

to obtain singlet excitation energy, we need to solve both singlet and triplet effective

Hamiltonian. The energy of a particular th singlet state is given by,

(1,1) (1,1) (1,1)( )S S S S

i eff ij jij

E C H C (12)

where, C and C are the left and right eigen vectors of the effective Hamiltonian.

7

B. Formulation of transition dipole moment

A general formalism to calculate transition dipole moments within the generalized

coupled cluster framework has been given Stolarczyk and Monkhorst.37

We follow the same

general formalism and extend it in the FSMRCC formalism.

For any Hermitian operator- O ; representing a certain perturbation to the system under

consideration, the perturbed Hamiltonian is given by –

ˆˆ ˆ( )H H O (13)

and the quantity

ˆ ˆq

p p qO O (14)

is the expectation value of the operator O , in the state p when, p = q. The above quantity is the

transition moment of O for the states p and

q when p q . The states p and

q are ortho-

normal Eigen states of the Hamiltonian. Due to the non-Hermitian nature of the normal coupled

cluster theory, the transition dipole moments are defined as the geometric mean of ˆ p

qO and ˆ p

qO .

Hence, transition dipole moment is defined as,

ˆ ˆpq p q q pd (15)

In the present paper we consider the electronic transition dipole moment (ETDM) from the

ground state to a few excited states. Hence,

†

p gr HF gr and

p gr gr HF

(16)

where † †(0,0)exp( )gr T

and

(0,0)exp( )gr T

The excited state wave function is given by,

8

(1,1)

(0)q ex ex and (1,1) †

(0)q ex ex (17)

Where, (0,0) (0,1) (1,0) (1,1)exp( ).ex T T T T The †

ex wave operator is the complex conjugate

of the ex wave operator. The excited state wave function, q is generated by the action of the

wave operator ex on the model space wave function given by,

(1,1) (1,1) (1,1)

(0) i ii

C (18)

The ground and excited state wave functions p and q are not normalized, in this method. If

we choose the operator O to be the one-electron dipole moment operator; then the transition

dipole moment for the ground to an excited state will be given by the following matrix element,

1

2† (1,1) (1,1) (1,1) (1,1) †ˆ ˆpq HF gr ex ex gr HFd O C C O

(19)

for a particular th state. There is a drawback of this expectation value formulation. The above

equation leads to a non-terminating series. For the practical application of evaluating transition

dipole moments, we have truncated it at the cubic level, under the CCSD approximation. In order

to give the expression a natural truncation, we invoke the bi-orthogonal approach put forward by

Jorgensen and co-workers.42

Using an extra set of de-excitation amplitudes the energy functional

is written as:

0

0 00

, 1 T T

o

T T T T

o q qq

F t e He

e He e He

(20)

where, '

q s are the de-excitation amplitude parameters of the conjugate ground state.

To calculate the first order property, we replace the Hamiltonian in the above expression with its

explicit first derivative and solve the above set of equations. It is worthwhile to point out, that we

can arrive at the above set of equations by linearizing the left vector of the extended coupled

cluster (ECC)43-44

functional also. The formalism developed by Jorgensen and co-workers is

9

known as the constrained variational approach (CVA). The CVA includes the z-vector45

method

as a zeroth order result.

The excited state wave function ex is represented in the above described constrained variation

approach as:

(1,1) (1,1) 11ex exC (21)

where, (0,0) (0,1) (1,0) (1,1) (22)

The Λ-amplitudes are decoupled in a manner opposite to that of the T-amplitudes. Hence, the

Λ(1,1)

amplitudes are solved first, followed by the Λ(0,1)

, Λ(1,0)

and finally the Λ(0,0)

amplitudes.

The details of this constrained variation approach is given in reference 41 and references therein.

The electronic transition dipole moment can now be evaluated from the following expression-

1

21 (1,1) (1,1) (1,1) (1,1) 1ˆ ˆ(1 ) 1pq HF gr ex ex gr HFd O C C O

(23)

Where O is the one-electron dipole moment operator. It may be pertinent to mention that,

although one body (1,1)T ( i.e. (1,1)

1T ) operator is formally there in the excited state description, it

has not been incorporated in the dipole moment matrix elements (19 or 23). This ETDM is not an

experimental observable. It is the square of this moment that is related to oscillator strength. So,

we calculate the dipole strength which is the square of the transition moment,

Dipole strength = 2

pqd (24)

And finally relate it to the oscillator strength which is given by,

22

3pq pqf E d (25)

where all quantities are in atomic units. The excitation energy, E is calculated from the

previously described FSMRCC method.

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III. COMPUTATIONAL ASPECTS

We have tested our method against various molecules like: CH+, BH, H2O and H2CO. To

test the accuracy of electronic transition dipole moments obtained from the first approach

mentioned earlier (see equation 19), we chose the CH+ molecule and BH molecule. A

comparison between the first approach (equation 19) and the second bi-orthogonal approach

(equation 23), is made for water and formaldehyde molecules. All the calculations carried out are

for singlet states. The ground state of all the molecules is treated, as a single reference coupled

cluster wave function and is solved in the manner as explained in section II. We denote the two

methods to evaluate transition moments with different abbreviations. The first method is denoted

as FSCC-T (refer equation 19). The second approach (refer equation 23) is denoted as FSCC- .

The Hartree-Fock determinant for the ground state is assumed to be the reference

function, which is treated as a vacuum for the Fock-space calculations. The model space is

formed by subsequent addition and/or removal of electrons to/from certain orbitals known as

active orbitals. The various Fock-space sectors and model space is represented in the particle-

hole formalism. An effective Hamiltonian is constructed whose diagonalization imparts the

energies of the corresponding states. The excitation energies are obtained directly as the energy

difference of the two states of choice. In the solution of Bloch equations, H is constructed as

T

CHe . This H is then contracted with Fock-space cluster amplitudes. Within CCSD

approximation, H is truncated up to three body terms. For the excited state wave function, we

have chosen a set of single active hole-particle (1,1) determinant as the model space. We have

used GAMESS46

to obtain the two-electron integrals. During the entire set of calculations, we

have not frozen any of the occupied or virtual core orbitals. Once the amplitudes are generated,

we calculate the transition dipole moments from the matrix elements as mentioned in equations

19 and 23.

Excited states of all the molecules were treated at the equilibrium geometry. Hence the

transition moments are calculated under the Frank Condon principle of fixed nuclear co-

ordinates. The calculated excitation energies (EE) are the vertical EE. These calculations scale as

N6. A comparison of FSCC-T and FSCC- is presented, against EOMCC method for the water

and formaldehyde molecules. The EOMCC results were obtained from ACES-II47

software

11

package. In order to test the size-intensivity, we have studied the variation of transition moments

from the FSCC-T and FSCC- approaches for the water monomer, water dimer and water

trimer at non-interacting distance. This is discussed in section IV-C and presented in Table IV.

IV. RESULTS AND DISCUSSIONS

A. CH+ Molecule

The CH+ molecule was chosen as a test molecule because extensive results were

available from other ab-initio calculations. The ground state electronic configuration of CH+ is

1σ22σ

23σ

2 which is chosen as vacuum. There is a large non-dynamical correlation in the ground

electronic state itself, arising from the interaction of 1σ22σ

23σ

2 and 1σ

22σ

21π

2 electronic

configurations. Due to this configuration mixing, some of the low lying states will have

appreciable double excitation character. Reference 30 lists the approximate excitation levels

(AEL) for some of the low-lying states of CH+. It was shown, that the state with the excitation

energy close to 3.2 eV is dominated by single hole-particle excited determinants within a set of

active orbitals- 3σ and 1п. We report transition moment for this particular transition and some

other excited states dominated by the single hole-particle excitation.

In the present calculation, the inter-nuclear distance was taken to be 2.13713a.u. The

calculations were performed with the basis set as given in the reference 30. We chose this

particular basis because full configurational-interaction (FCI) and other theoretical results were

available for this basis. A split valence basis, augmented with two diffuse s and p functions and

one d polarization function was used for the carbon atom. The hydrogen atom was augmented

with one diffuse ‘s’ function and one ‘p’ polarization function. Table I presents the transition

energies, electronic transition dipole moment, dipole strength and the oscillator strength of CH+

molecule in the basis mentioned above.

On comparing the reported values in Table I, we find that the transition moment,

transition energy and hence the oscillator strength values (within the given basis) as obtained

from FSCC-T method are close to the FCI results. The Fock-space active space that we have

chosen is 4σ2π (one active hole and seven active particles). We have also reported FSMRCC

12

results- as calculated and tabulated by Barysz,39

following the formulation of Stolarczyk and

Monkhorst 37

for the 3σ to 1п transition.

A comparison with the EOMCC method has been made for the transitions arising from

3σ to 4σ and 3σ to 5σ. FSCC-T transition moment agrees well with the EOMCC transition

moment for the higher excitations as well. The EOMCC result reported for these transitions has

been obtained from ACES-II47

package. Transition dipole moments have also been reported for

transitions arising from the 3σ to 6σ state.

B. BH Molecule

In general, boron hydrides are a fundamentally important group of compounds for

synthetic and theoretical chemical studies. The smallest but stable boron hydride is B2H6. It is a

well studied compound. However, some of the intermediate species are not well studied due to

their high reactivity. One of these intermediates is BH. This particular boron hydride is one of

the simplest molecules that can easily be studied under ab-initio theories.

In the present calculation, we treat the BH molecule at the equilibrium distance of 2.3289

atomic units. The BH molecule is iso-electronic with the CH+ molecule. The ground state of BH

is of A1Σ

+ symmetry, while the first excited state is of X

1Π symmetry. It is a six electron system

and its electronic configuration is given by 1σ22σ

23σ

2, which is chosen to be the vacuum. The

single reference ground state wave function is generated by the action of the cluster operator on

this reference space. We chose our model space such that, 3σ is the active hole. 7 active particles

were chosen which include three σ symmetric orbitals and two п type orbitals. Hence, the total

model space span four σ symmetric and two п symmetric orbitals (1 active hole and 7 active

particles). We have reported the excitation energies, transition moments as well as the oscillator

strength of BH molecule at the aforesaid geometry in three different basis sets for a few low-

lying excited states. We report ETDM values in cc-pVDZ, augmented cc-pVDZ and augmented

cc-pVTZ basis sets.

Table II(a) show the comparison of our method with EOMCC results in cc-pVDZ basis.

In this basis, the electronic transition dipole moment and hence the oscillator strength value

13

agrees well with the EOMCC method for all the reported excitations (as mentioned in Table IIa).

The difference in transition moments is slightly pronounced for the 3σ to 4σ transition as

compared to the other two transitions. The excitation energy is also seen to differ more for this

particular transition. The combined effect of this is seen in the oscillator strength value, which

differs from the EOMCC value by about 0.04 atomic units. The excitation energies for the two

other transitions agree with the EOMCC method. Hence, the oscillator strength as obtained from

FSCC-T is comparable to that obtained from EOMCC.

On introducing augmentation, the transition energies are seen to decrease. In the

augmented basis sets, we have reported transitions from the 3σ to 1п, 4σ and 2п states

respectively. In the aug-cc-pVDZ basis (refer Table II(b)), the oscillator strengths as calculated

from FSCC-T method, from 3σ to both the п-symmetry states agree better with the EOMCC

results as compared to the 3σ to 4σ transition. On moving from the augmented double zeta to

augmented triple zeta basis certain points are noteworthy. The excitation energy for the 3σ to 1п

state remains constant for the FSMRCC method. On the other hand, the excitation energies for

the 3σ to 4σ and 2п states remain constant for the EOMCC method. The oscillator strengths of

all the reported excitations, remains constant for the EOMCC method on moving from double

zeta to triple zeta basis. While in FSCC-T method, the transition moments change. This cause the

oscillator strength to change as well, even though the excitation energy remains constant for one

of the transitions. This brings about a difference in oscillator strength between the two methods

in the augmented triple zeta basis.

C. H2O Molecule

Various calculations on the water molecule were performed at the ground state

equilibrium geometry. A detailed comparison of the method developed by us and other

theoretical methods available is presented for the water molecule. Table III presents the results in

both FSCC-T and FSCC- methods. We have tabulated results in three different basis sets for a

number of transitions.

For water, the ground state restricted Hartree-Fock determinant is chosen as the vacuum which is

given by,

14

2 2 2 2 2

1 1 1 1 21 2 1 3 1HF a a b a b (26)

where, 1b2 is the highest occupied orbital. The first unoccupied orbital is of a1 symmetry. We

chose our model space such that 1b1,3a1 and 1b2 are the active holes and 4a1 and 2b1 are the

active particles for cc-pVTZ and cc-pVQZ basis. In case of augmented basis, our model space

includes two active holes and six active particles.

Table III(a) and III(b) report calculations performed on correlation consistent basis sets,

developed by Dunning.48

We present the results for the FSCC-T as well as FSCC- in both the

basis sets. Both our methods, FSCC- and FSCC-T agree well with the calculated EOMCC

values. The FSCC-T method and FSCC- method differs in the transition moment values by

about 0.05 atomic units to 0.08 atomic units. This is due to the manner in which the wave

function has been formulated and constructed. Transition energies for all the reported excitations

are matching well with the corresponding EOMCC transition energies. In Table III(a), the

transition dipole moment of state-3 match exactly with that of EOMCC. This leads to a fairly

good agreement of the oscillator strength of the two methods. But, barring this particular state,

the transition dipole moment value obtained from the EOMCC method lies between the FSCC-T

and FSCC- methods.

On increasing the valence triple zeta to valence quadruple zeta, the excitation energies of

all the reported excitations decreases. This trend is seen for both the FSMRCC and EOMCC

methods. The transition dipole moment also shows a lowering of its value. This general trend

among the transitions is not shown by the transition represented by state-3. In this particular

state, the transition dipole moment is seen to increase on increase in valence zeta basis. Even

though the excitation energy for this particular transition shows a lowering of value, the

oscillator strength, being a product of dipole strength and transition energy, show an increment.

The experimental oscillator strength value of 0.041a.u.49

is reported for the specific transition as

reported in state-1. The convergence toward experimental value is seen on moving from the

triple zeta to the quadruple zeta basis.

We have also calculated transition moments and oscillator strengths in augmented cc-

pVTZ basis. In case of water, all of the excited states have a fair degree of Rydberg character and

hence introduction of diffuse functions in the basis set lowers the excitation energy of the 1b2 to

15

4a1 state. In this particular basis, we chose 2 active holes and 6 active particles as our model

space. In Table III(c), we present transition moments for four excited states of water in this basis.

A comparison is made with EOMCC wherever applicable. We compare the transition dipole

moment values for FSCC-T and FSCC-Λ methods. The FSCC-T and FSCC- transition

moment values differ amongst themselves by 0.09a.u. to 0.001a.u. Since, the excited states

described by the augmented basis is different from those described by the previous basis sets, no

concrete conclusion can be drawn from the other low lying excited states other than the one

described in state-1 (which is the homo-lumo transition). The qualitative trend of lowering of

excitation energy is noteworthy for this HOMO-LUMO transition. Though the excitation energy

decreases, the transition dipole moment show considerable increment and hence affects

(increases) the oscillator strength for this particular state.

We have tested both FSCC-T and FSCC- formulations, to check whether the transition

dipole moments are size-intensive or size-extensive. We have calculated the transition dipoles

for water monomer, dimer and trimer at non-interacting distances in FSCC-T and FSCC-

formulations. The water molecules are treated at the equilibrium ground state geometry in cc-

pVZD basis. The results are presented in Table IV. We find that the transition moments are size-

intensive in both the FSCC methods as the transition dipole remains constant with increase in

water monomer unit.

D. H2CO Molecule

As a final check for the developed FSCC-T and FSCC- methods, we chose the

formaldehyde molecule. Table V present the excitation energies, transition moments, dipole

strengths and oscillator strengths for nine transitions in cc-pVDZ basis. Together with the results

obtained from both the FSCC approaches, we also report values obtained from EOMCC method,

for comparison. The C-O and C-H bond distance was taken to be 1.20838Å and 1.116351Å. The

H-C=O bond angle is 121.75 degrees.

For formaldehyde, the ground state restricted Hartree-Fock determinant is chosen as the vacuum

which is given by,

16

1 1 1 1 2 1 1 21 2 3 4 1 5 1 2HF a a a a b a b b

We chose four active holes and four active particles as the model space. Hence, 1b2, 5a1, 1b1 and

2b2 are chosen as active holes. 2b1, 6a1, 3b2 and 7a1 are chosen as the active particles. The FSCC-

T method of evaluating transition dipole moments, generate higher values as compared to the

transition moment obtained from FSCC- method in almost all the reported transitions. In

certain transitions, the EOMCC method evaluate transition dipoles that agree very well with

FSCC- , while in some transitions it agrees better with FSCC-T formulation. Both FSCC-T and

FSCC- generate transition moments that are comparable with each other.

V. CONCLUDING REMARKS

In this article, we have described two ways to evaluate electronic transition dipole

moments within the Fock-space multi-reference coupled cluster framework. We report transition

energies, transition moments and oscillator strengths for CH+, BH, H2O and H2CO molecules.

We observe that both FSCC-T and FSCC- provide transition dipole moments that are in close

proximity of each other. Also, both these methods agree well with EOMCC transition moments.

We also conclude that the transition moments formulated in this Fock-space formalism,

FSCC-T and FSCC- are both size-intensive. This is in contrast to EOMCC, where the left

transition moment is not size-intensive, while the right transition moment is size-intensive.

VI. ACKNOWLEDGEMENT

The authors acknowledge the facilities provided by the centre of excellence in scientific

computing present at CSIR National Chemical Laboratory (NCL). Two of the authors SP and

DB acknowledge the grant sanctioned from the Department of Science and Technology (DST),

through J.C. Bose fellowship project for financial support.

17

References:

1 G. Scholes, Annu. Rev. Phys. Chem. 54, 57 (2003)

2 D. Toptygin, J. Fluores. 13, 201 (2003)

3 See, L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), Chaps.

11 and 14.

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19

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20

Table I: Transition energies, transition moments, dipole strengths and oscillator strengths of CH+

molecule from its ground state to a few excited states are tabulated. EE stands for excitation

energy. TDM is transition dipole moments, DS is dipole strength and OS is the oscillator

strength. All reported values are in atomic units. The basis set and geometry is given in text.

CH+

State-1 State-2 State-3 State-4

FSCC-Ta EOMCC

b FCI

c FSCC

d FSCC-T

a EOMCC

b FSCC-T

a EOMCC

b FSCC-T

a

EE 0.1196 0.1198 0.1187 0.1191 0.5067 0.4990 0.6571 0.6544 0.6835

TDM 0.296 0.306 0.299 0.243 0.971 1.036 0.198 0.176 1.293

DS 0.088 0.095 0.089 0.059 0.943 1.073 0.039 0.031 1.672

OS 0.0069 0.0076 0.0070 0.0046 0.3187 0.3571 0.0173 0.0135 0.7618

a Our Method(refer equation 19),

b Obtained from ACES-II package, see reference 47,

c see

reference 30 , d

see reference 39

State 1: 3 1 , State 2: 3 4 , State 3: 3 5 , State 4: 3 6

21

Table II (a): Transition energies, transition moments, dipole strengths and oscillator strengths of

a few excited states of BH molecule in cc-pVDZ basis are presented in this table. EE is the

excitation energy. TDM is transition dipole moments, DS is dipole strength and OS is the

oscillator strength. All reported values are in atomic units. The equilibrium bond distance is

2.32899a.u.

cc-pVDZ basis State -1 State -2 State - 3

EE FSMRCC 0.1114 0.3670 0.4757

EOMCCa 0.1119 0.3647 0.4746

TDM FSCC-T 0.598 1.651 0.405

EOMCCa 0.613 1.700 0.399

DS FSCC-T 0.358 2.726 0.164

EOMCCa 0.376 2.891 0.159

OS FSCC-T 0.0266 0.6668 0.0520

EOMCCa 0.0281 0.7030 0.0504

State – 1: 3 1 , State – 2 : 3 4 , State – 3 : 3 5

a Obtained from ACES-II package, see reference 47.

22

Table II (b): Transition energies, transition moments, dipole strengths and oscillator strengths of

a few excited states of BH molecule in augmented cc-pVDZ basis are reported. EE is the

excitation energy, TDM is transition dipole moments, DS is dipole strength and OS is the

oscillator strength. The reported values are in atomic units. The equilibrium bond distance is

2.32899a.u.

aug-cc-pVDZ State -1 State -2 State - 3

EE FSMRCC 0.1085 0.2423 0.2793

EOMCCa 0.1091 0.2407 0.2811

TDM FSCC-T 0.588 0.843 0.313

EOMCCa 0.588 0.836 0.314

DS FSCC-T 0.345 0.711 0.098

EOMCCa 0.346 0.699 0.098

OS FSCC-T 0.0249 0.1148 0.0183

EOMCCa 0.0251 0.1122 0.0185

State – 1: 3 1 , State – 2 : 3 4 , State – 3 : 3 2

a Obtained from ACES-II package, see reference 47.

23

Table II (c): Transition energies, transition moments, dipole strengths and oscillator strengths of

a few excited states of BH molecule in augmented cc-pVTZ basis are presented. EE is the

excitation energy, TDM is transition dipole moments, DS is dipole strength and OS is the

oscillator strength. The reported values are in atomic units. The equilibrium bond distance is

taken to be 2.32899a.u.

aug-cc-pVTZ State -1 State -2 State - 3

EE FSMRCC 0.1085 0.2473 0.2849

EOMCCa 0.1071 0.2407 0.2812

TDM FSCC-T 0.575 0.933 0.335

EOMCCa 0.592 0.836 0.314

DS FSCC-T 0.330 0.871 0.112

EOMCCa 0.350 0.699 0.099

OS FSCC-T 0.0239 0.1437 0.0214

EOMCCa 0.0250 0.1122 0.0185

State – 1: 3 1 , State – 2: 3 4 , State – 3: 3 2

a Obtained from ACES-II package, see reference 47.

24

Table III(a): Excitation energies, transition moments, dipole strengths and oscillator strengths for

water molecule in cc-pVTZ basis is presented.EE is the excitation energy, TDM is transition

dipole moments, DS is dipole strength and OS is oscillator strength. All the reported values are

in atomic units. The calculations were performed at the experimental ground state geometry, r =

0.957 and θ = 104.5˚.

cc-pVTZ basis State-1 State-2 State-3 State-4 State-5

EE FSMRCC 0.2952 0.6321 0.3864 0.4649 0.5310

EOMCCa 0.2964 0.6302 0.3879 0.4652 0.5317

TDM

FSCC-T 0.444 0.446 0.684 0.468 0.826

FSCC- 0.396 0.394 0.637 0.382 0.777

EOMCCa 0.421 0.435 0.637 0.439 0.806

DS

FSCC-T 0.197 0.199 0.468 0.219 0.683

FSCC- 0.157 0.156 0.406 0.146 0.604

EOMCCa 0.177 0.189 0.406 0.192 0.650

OS

FSCC-T 0.0389 0.0841 0.1206 0.0678 0.2417

FSCC- 0.0309 0.0657 0.1047 0.0453 0.2138

EOMCCa 0.0351 0.0795 0.1050 0.0597 0.2304

Experimental Oscillator strength49

for the transition represented by State-1: 0.041a.u.

a Obtained from ACES-II package, see reference 47

State 1: 2 11 4b a , State 2: 1 11 2b b , State 3: 1 13 4a a , State 4: 1 13 2a b , State 5: 1 11 4b a

25

Table III(b): Excitation energies, transition moments, dipole strengths and oscillator strengths for

water molecule in cc-pVQZ basis is presented. EE is the excitation energy, TDM is transition

dipole moments, DS is dipole strength and OS is oscillator strength. All the reported values are

in atomic units.

cc-pVQZ basis State-1 State-2 State-3 State-4 State-5

EE FSMRCC 0.2938 0.6181 0.3826 0.4593 0.5271

EOMCCa 0.2934 0.6070 0.3827 0.4574 0.5260

TDM

FSCC-T 0.502 0.383 0.714 0.422 0.786

FSCC- 0.426 0.320 0.670 0.374 0.696

EOMCCa 0.465 0.322 0.650 0.397 0.749

DS

FSCC-T 0.252 0.146 0.510 0.178 0.617

FSCC- 0.182 0.102 0.450 0.139 0.484

EOMCCa 0.216 0.104 0.422 0.157 0.561

OS

FSCC-T 0.0493 0.0604 0.1301 0.0546 0.2169

FSCC- 0.0356 0.0423 0.1147 0.0427 0.1701

EOMCCa 0.0424 0.0419 0.1077 0.0480 0.1966

Experimental Oscillator strength49

for the transition represented by State-1: 0.041a.u.

a Obtained from ACES-II package, see reference 47.

State 1: 2 11 4b a , State 2: 1 11 2b b , State 3: 1 13 4a a , State 4: 1 13 2a b , State 5: 1 11 4b a

26

Table III(c): Excitation energies, transition moments, dipole strengths and oscillator strengths in

augmented cc-pVTZ basis for water molecule is reported. EE is the excitation energy, TDM is

transition dipole moments, DS is dipole strength and OS is oscillator strength. All the reported

values are in atomic units.

aug-cc-pVTZ basis State-1 State-2 State-3 State-4

EE FSMRCC 0.2796 0.3655 0.4180 0.5072

EOMCCa 0.2802 0.3663 - -

TDM

FSCC-T 0.575 0.702 0.054 0.307

FSCC- 0.494 0.638 0.045 0.283

EOMCCa 0.542 0.633 - -

DS

FSCC-T 0.331 0.492 0.003 0.094

FSCC- 0.244 0.407 0.002 0.080

EOMCCa 0.294 0.401 - -

OS

FSCC-T 0.0616 0.1200 0.0008 0.0319

FSCC- 0.0455 0.0992 0.0006 0.0272

EOMCCa 0.0549 0.0978 - -

Experimental Oscillator strength49

for the transition represented by State-1: 0.041a.u.

a Obtained from ACES-II package, see reference 47.

State 1: 2 11 4b a , State 2: 1 13 4a a , State 3: 2 21 2b b , State 4: 1 13 6a a

27

Table IV: Water monomer, dimer and trimer in cc-pVDZ basis. Experimental ground state

geometry is, r = 0.957 and θ = 104.5˚. The monomer units were placed at non-interacting

distance to check the size-intensivity of the FSCC-T and FSCC-Λ methods. TDM stands for

transition dipole moments.

cc-pVDZ basis H2O monomer H2O Dimer H2O Trimer

TDM

FSCC-T 0.339 0.339 0.339

FSCC- 0.297 0.297 0.297

28

Table V: Excitation energies, transition moments, dipole strengths and oscillator strengths in

augmented cc-pVDZ basis for the formaldehyde molecule is tabulated. EE is the excitation

energy, TDM is transition dipole moments, DS is dipole strength and OS is oscillator strength.

All the reported values are in atomic units. The C-O and C-H bond distances are 1.20838Å and

1.116351Å. The H-C=O bond angle is 121.75 degrees.

cc-pVDZ basis State1 State2 State3 State4 State5 State6 State7 State8 State9

EE FSCC 0.3106 0.4115 0.4194 0.5689 0.3415 0.5149 0.6252 0.5409 0.6743

EOMCCa 0.3145 0.4180 0.4235 - 0.3494 0.5197 0.6268 0.5476 -

TDM

FSCC-T 0.849 1.482 0.299 0.528 0.125 0.567 0.886 1.100 0.580

FSCC- 0.782 1.399 0.256 0.541 0.119 0.512 0.873 1.008 0.609

EOMCCa 0.813 1.338 0.295 - 0.091 0.505 0.876 0.997 -

DS

FSCC-T 0.722 2.197 0.089 0.278 0.016 0.321 0.785 1.210 0.337

FSCC- 0.612 1.958 0.065 0.292 0.014 0.262 0.762 1.016 0.371

EOMCCa 0.661 1.791 0.086 - 0.008 0.255 0.768 0.994 -

OS

FSCC-T 0.1494 0.6029 0.0250 0.1056 0.0036 0.1102 0.3274 0.4363 0.1517

FSCC- 0.1267 0.5372 0.0183 0.1108 0.0032 0.0900 0.3176 0.3664 0.1670

EOMCCa 0.1385 0.4993 0.0245 - 0.0019 0.0885 0.3210 0.3632 -

State 1: 2 12 6b a , State 2: 2 22 3b b , State 3: 2 12 7b a , State 4: 1 11 7b a ,

State 5: 1 15 2a b , State 6: 1 15 6a a , State 7: 1 15 7a a , State 8: 2 11 6b a , State 9: 2 11 7b a

a Obtained from ACES-II package, see reference 47.