1 Introduction

Development of efficient and accurate quantum mechanical method for the description of anharmonic vibrations in polyatomic molecules has been a key interest of many researchers. Within the Born-Oppenheimer approximation, the electronic Schrödinger equation generates the potential energy surface for the nuclear motion in the molecule. The Watson Hamiltonian describes the molecular vibrations in a simple and efficient way

$$\begin{aligned} H=\sum _i\frac{P_i^2}{2}+V(q)+V_W+V_C. \end{aligned}$$
(1)

Here, \(q_i\) are the mass-weighted normal coordinates and \(P_i\) are the conjugate momenta. The terms \(V_W\) and \(V_C\) are the Watson’s mass-dependent term and Coriolis coupling term, respectively. The potential V(q) is usually approximated by the quartic polynomial of the Taylor series expansion

$$\begin{aligned} V(q) =\frac{1}{2}\sum _i\omega _i^2q_i^2 + \sum _{i\le j\le k}f_{ijk}q_iq_jq_k + \sum _{i\le j\le k\le l}f_{ijkl}q_iq_jq_kq_l. \end{aligned}$$
(2)

Here, \(\omega _i\) is the harmonic frequency of \(\mathrm{i}^{th}\) vibrational mode, \(f_{ijk}\) and \(f_{ijkl}\) are the third and fourth derivatives of the electronic energy with respect to the mass-weighted normal coordinates at equilibrium geometry.

The vibrational Hamiltonian with such quartic potential is a many-body Hamiltonian, and thus, the exact analytical solution of corresponding Schrödinger equation is not possible. Several approximate methods have been developed to solve the vibrational Schrödinger equation based on both variation principle and perturbation theory. The second order vibrational perturbation theory (VPT21,2,3) has been developed, where the Hamiltonian with only quadratic potential is taken as the zeroth order Hamiltonian and the cubic and quartic terms are treated as perturbations. Although, such approach has been used extensively by many authors with successful interpretations of infra-red spectra of numerous molecules, the failure of the VPT2 is well-known when the one encounters vibrational resonances like Fermi resonance. The vibrational self-consistent field (VSCF) theory4,5,6 and its generalizations to multiconfigurational reference functions7 are developed and used extensively to describe the experimental IR spectra of molecules with large number of vibrational modes. The vibrational configuration interaction (VCI)8,9,10 method is also developed and applied to small molecules. Here, one constructs and diagonalizes the vibrational Hamiltonian matrix in a configuration space generated by VSCF virtual orbitals, or distributed Gaussian functions. The VCI method is in principle exact. However, the dimension of the Hamiltonian matrix increases exponentially with the vibrational degrees of freedom, making it inapplicable for even medium size molecules.

The coupled cluster method11,12,13,14,15,16,17,18,19,20 has been emerged as the most successful method for describing the quantum many-body problems, especially for electronic structures of atoms and molecules. The formulation of the coupled cluster method is inherently non-variational and non-perturbative, where the exact ground state wave function is parametrized with an exponential wave operator operating on a reference wave-function, \(\vert 0\rangle \)

$$\begin{aligned} \vert \Psi _g \rangle = U \vert 0\rangle , \end{aligned}$$
(3)

with either

$$\begin{aligned} U&=e^{S} \end{aligned}$$
(4a)
$$\begin{aligned} {\text{ or, }} U&=e^Se^{-\sigma } \end{aligned}$$
(4b)

Here, S, and \(\sigma \) are the cluster operators, consist of connected one-body, two-body up to n-body excitation and de-excitation operators, respectively. The working equations to determine the cluster matrix elements are given by

$$\begin{aligned} \langle e\vert U^{-1}HU\vert 0\rangle = 0, \end{aligned}$$
(5)

where, \(\vert e\rangle \) are the excited states obtained by the excitation from the reference function \(\vert 0\rangle \). By virtue of the exponential wave operator, one can access the unlinked higher excited state functions, and as a result the energy and wave function are highly accurate even with a low order approximation of the cluster operator.

Two different representations have been developed to formulate the vibrational coupled cluster method (VCCM).21,22,23,24,25,26,27,28,29,30,31,32,33,34 The fundamental difference between these formulations is the choice of the reference function and definition of excitation/de-excitation operators. In one representation, a set of orthonormal basis function is assigned to each vibrational degree of freedom. The VSCF calculation is carried out using these basis set and subsequently, the VSCF ground state wave function is used as the vacuum. The creation and annihilation operators are defined in a manner similar to the coupled cluster theory of Fermionic systems. This approach is known as the basis set representation of VCCM.21,22,23,24 The accuracy of the results with this method depends on the size and quality of single mode basis functions along with the truncation of the cluster operator. This approach was used by Latha and Prasad to describe the intra-molecular vibrational relaxation dynamics,35 and Christiansen and coworkers21,22,23,24 to compute the IR spectra. The other representation is known as the bosonic representation, developed and implemented by Prasad and co-worker,25,26,27,28,29,30,31,32,34 and, more recently by Hirata.33 The vacuum here is an optimized multi-dimensional Gaussian product function and the cluster operator is constructed with simple harmonic oscillator (HO ladder operators that are defined with respect to the optimized vacuum state. In our earlier works, we used the bosonic representation to formulate the VCCM to calculate the vibrational excitation energies,26 infra-red27 and Raman intensities.34

The bosonic representation of VCCM offers several advantages over the basis set representation. Unlike the basis set representation, it is invariant to any arbitrary rotations and displacements in normal coordinate space. Secondly, since the excitation operators are the HO ladder operators, once can access, in principle, the full Hilbert space of the HO eigenfunctions. However, depending on the truncation of the cluster operator, the number of independent co-efficients in equation 5 is finite. In a previous study,26,27 we found that the convergence is quite rapid, essentially by \(S_4\), the ground state energy is converged.

Two different approaches have been developed to describe the excited state within the VCCM formulation in bosonic representation. One is the vibrational multi-reference coupled cluster method based on effective Hamiltonian formalism.28 The second approach is the coupled cluster linear response theory (CCLRT), variously known as the equation of motion coupled cluster (EOMCC) method.25,27,36 In this approach, one constructs a similarity transformed effective Hamiltonian

$$\begin{aligned} H_{eff} = U^{-1}HU, \end{aligned}$$
(6)

in the space of zeroth-order excited states and diagonalizes it to get the excitation energies directly.

The equations to determine the cluster matrix elements (equation 5) are coupled non-linear equations, and they are solved iteratively in a self-consistent manner. The solution of these equations is one of the major computational bottleneck. When a four-body approximation of cluster cluster operator is used, the computational cost scales up to \(N^6\) (here, N is the number of vibrational degrees of freedom) at each iteration. Using the diagrammatic arguments it is realized that the coupled cluster wave function and energy can be decomposed in terms of infinite order of perturbation theory terms of selected excitation operators. In this work, we explore the possibility of using a low order perturbative approximation to the cluster operators for the VCCM. In other words, the effective Hamiltonian in equation 6 for the CCLRT is constructed using only the first order estimate of two-body, three-body and four-body S and \(\sigma \) operators. The motivation for doing so comes from two sources. The computation of the vibrational spectra consists of three steps. In the first step, the coupled nonlinear equations of the VCCM are to be solved. In the second step, \(H_{eff}\) has to be constructed. Finally, the \(H_{eff}\) has to be diagonalized in the manifold of excited states. By invoking the perturbative expansion, the computational burden of the first step reduces from iterative \(N^6\) to a non-iterative \(N^4\) scaling of CPU time. For a large molecular system, this itself is a desirable feature and hence worth exploring. The second source of motivation comes from a desire to understand the relative importance of changes in correlation and relaxation during the excitation process. Approximation to the cluster operator affect the computation of the changes in the ground state correlation energy, while the excitation manifold controls the accuracy of the relaxation effects. Thus, a perturbative approximation can give an insight in to the relative importance of these two quantities. We note that, in the context of electronic structure theory, Nooijen and Snijders,37 Stanton and Gauss,38 Pal and co-workers,39 Bartlett and co-workers40 have explored the possibility of expanding the cluster operator, S, in a perturbative manner. Numerical studies indicate that such perturbative truncation of S, within the CCLRT framework for the excited sates, gives satisfactory results, even when the perturbative expansion is kept as low as the first order for S. Several studies attempted to incorporate the effects of higher rank cluster operators within in the mathematical framework of lower rank approximation (such as the influence of \(T_3\) operator with in the CCSD framework) by using perturbative approximations.41,42,43 Our own goal is different. We are not looking to simulate the influence of higher rank cluster operator. We seek to understand the importance of different terms in the open part of \(H_{eff}\) on the approximate spectrum from a perturbation theory perspective. To that end, as a first step, we approximate the cluster matrix elements by their first order perturbative estimates.

The goal of the present study is to test the accuracy of the vibrational energy levels using similar low order expansions to the cluster operator that is used to generate \(H_{eff}\). The theoretical framework for this is presented in Section 2. Section 3 contains the computational results on two molecules, water and 1,1-difluoroethylene. The results indicates that perturbative expansion of S, correct up to first order, gives small errors compared to the full VCCM approach, about 5–10 cm\(^{-1}\), while the computational cost is significantly reduced.

2 Theory

2.1 Coupled Cluster Method in bosonic representation

In the first step of the VCCM formalism, we invoke the effective harmonic oscillator (EHO) approximation,25,44 to get an optimized reference state for the vibrational ground state. In EHO method, a product of N Gaussian functions

$$\begin{aligned} \vert 0\rangle =exp\left( -\sum _{i}\omega _{i}(q_{i}-q_{i}^0)^2/2\right) , \end{aligned}$$
(7)

is variationally optimized with respect to \(\omega _i\) and \(q_i^0\). Here, \(\omega _i\) and \(q_i^0\) are the harmonic frequency of the normal mode and origin of the normal coordinate i, respectively. The harmonic oscillator creation and annihilation operators are then defined with respect to this optimized state as

$$\begin{aligned} a_{i}= & {} \sqrt{\frac{\omega _{i}}{2}}\left( q_{i}-q_{i}^0 +\frac{1}{\omega _{i}}\frac{d}{dq_{i}}\right) \end{aligned}$$
(8)
$$\begin{aligned} a_{i}^\dagger= & {} \sqrt{\frac{\omega _{i}}{2}} \left( q_{i}-q_{i}^0-\frac{1}{\omega _{i}}\frac{d}{dq_{i}}\right) \end{aligned}$$
(9)

The optimized function in equation 7 is, by definition, a vacuum state for the annihilation operator \(a_i\) as it satisfy the killer condition

$$\begin{aligned} a_{i}\vert {0}\rangle = 0. \end{aligned}$$
(10)

Unlike the traditional single reference coupled cluster formalism, where only the excitation operators are use to define the wave operator, we used both excitation and de-excitation cluster operators to parametrize the ground state wave function in the VCCM formalism in the spirit of extended coupled cluster method (ECCM) of Arponen.45,46,47 However, we do not follow Arponen’s bi-variational approach to determine the cluster matrix elements, as discussed below. The use of ECCM ansatz offers several advantages over the traditional coupled cluster formalism. First, the ground state bra vector is also exponentially parametrized along with the ground state ket vector in the ECCM approach. The exponentially parametrized bra vector for the ground state gives a balanced description for the transition matrix elements.27,32,36 Secondly, the excitation energies are obtained by the diagonalization of similarity transformed effective Hamiltonian within the CCLRT framework. The similarity transformed effective Hamiltonian is manifestly non-hermitian. Diagonalization of non-hermitian effective Hamiltonian occasionally generates spurious complex eigenvalues, when a low-order truncation of cluster operator S is used for the construction of \(H_{eff}\) matrix. In the ECCM, the effective Hamiltonian is now a double-similarity transformed Hamiltonian, with a de-excitation operator used for the second similarity transformation. Since the de-excitation operators are the adjoint of the excitation operators, the second similarity transformation eliminates substantial parts of the non-hermitian residue in the effective Hamiltonian. The reduction of the non-hermicity due to the second similarity transformation gives a better description for the excited states, eliminating some of the spurious complex eigenvalues.26

The ground state wave function for VCCM is thus written as,

$$\begin{aligned} \vert \Psi _g\rangle = e^Se^{-\sigma }\vert 0\rangle . \end{aligned}$$
(11)

Here, the cluster operators S and \(\sigma \) are consisting of one quanta, two quanta, etc., excitation and de-excitations operators, respectively.

$$\begin{aligned} S&= \sum _is_ia_i^\dagger + \sum _{i\le j}s_{ij}a_i^\dagger a_j^\dagger +\sum _{i\le j\le k}s_{ijk}a_i^\dagger a_j^\dagger a_k^\dagger + \cdots \nonumber \\ \sigma&= \sum _i\sigma _i a_i + \sum _{i\le j}\sigma _{ij}a_i a_j+\sum _{i\le j \le k}\sigma _{ijk}a_ia_ja_k+\cdots \end{aligned}$$
(12)

The working equations for cluster operator S, ground state energy \(E_g\), and cluster operator \(\sigma \) are obtained from,

$$\begin{aligned}&\langle e\vert e^{-S}He^S\vert 0\rangle =0, \end{aligned}$$
(13)
$$\begin{aligned}&E_g=\langle 0\vert e^{-S}He^S\vert 0\rangle , \end{aligned}$$
(14)
$$\begin{aligned}&\langle 0\vert e^{\sigma }e^{-S}He^Se^{-\sigma }\vert e\rangle =0, \end{aligned}$$
(15)

respectively. Note that, the original formulation of ECCM by Arponen invokes a variational solution for the cluster matrix elements \(\sigma \).45,46,47 However, we use the method of moment approach of equation 15 in the spirit of conventional coupled cluster solutions, as suggested by Prasad,48 and implemented in several of our earlier studies.26,27,28,29,30,31,34 Within this framework, the equations for S are decoupled from the equations for \(\sigma \). Such decoupling can be shown to be exact in the exact limit of both S and \(\sigma \) using Lie algebraic decoupling theory,48 and by looking at the coupled cluster method as a similarity transformed effective Hamiltonian theory and relating the eigenfunctions of the Hamiltonian to the left and right eigenfunctions of the transformed Hamiltonian. Since these two vectors are constructed separately, the left eigenvector (represented by \(e^{-\sigma }\)) does not affect the construction of the right eigenvector (\(e^S\) by construction). However, it carries the burden of normalization of the coupled cluster ansatz for the ket state.36 Hence, \(\sigma \) depends on S, but not vice versa. Both equations 13 and 15 are sets of coupled non-linear equations that are solved consecutively in a self-consistent manner to get converged sets of S and \(\sigma \) matrix elements respectively. The ground state energy is independent of \(\sigma \) matrix elements. Thus, the second similarity transformation keeps the ground state unaltered, but affects the description of the excited states. The similarity transformed Hamiltonians of equations 13 and 15 are written in terms of Hausdorff expansion,

$$\begin{aligned} H_{eff}^1= e^{-S}He^S=H+[H,S]+\frac{1}{2!}[[H,S],S]\nonumber \\&+\frac{1}{3!}[[[H,S],S],S]+\cdots , \end{aligned}$$
(16)

and,

$$\begin{aligned} H_{eff}^2= e^\sigma H_{eff}^1e^{-\sigma }=H_{eff}^1+[\sigma ,H_{eff}^1]+ \frac{1}{2!}[\sigma ,[\sigma ,H_{eff}^1]] \nonumber \\&+\frac{1}{3!}[\sigma ,[\sigma ,[\sigma ,H_{eff}^1]]]+\cdots \end{aligned}$$
(17)

The commutator structure of the above expressions ensure that diagrammatically only the connected terms will survive. Since the Hamiltonian has a maximum four-body operator, the series in equation 16 terminates after fourth commutator term, irrespective of the truncation used for the S cluster operator. Note that the operators in the effective Hamiltonian \(H_{eff}^1\) are many more and the number depends on the truncation of cluster operator S. For example, for a four-body truncation of S, the maximum number of operators in \(H_{eff}^1\) is twelve. Similar situation holds for the double-similarity transformed effective Hamiltonian \(H_{eff}^2\). In the implementation of VCCM, both \(H_{eff}^1\) and \(H_{eff}^2\) are approximated with maximum of four boson operators.

The excited states are described using CCLRT. Here, the excited state wave functions are obtained by the action of a linear operator on the ground state CCM wave function

$$\begin{aligned} \Psi _e = X_e U\vert 0\rangle . \end{aligned}$$
(18)

Generally, \(X_e\) requires both excitations and de-excitation operators to define it both from operator completeness requirements,49,50,51 and the physical requirement to describe the population changes in various zeroth order states from ground state to the excited state. However, in the present case, it is not necessary.

Equation 18 can be rewritten as

$$\begin{aligned} \vert \Psi _e\rangle = UU^{-1}X_e U\vert 0\rangle = UX_e^{eff}\vert 0\rangle . \end{aligned}$$
(19)

The operator \(X_e^{eff}\) now acts on the vacuum, and hence requires excitation operators only. The general expression for the \(X_e^{eff}\) can be written as,

$$\begin{aligned} X_e^{eff} = \sum _iX_{e,i}^{eff}a_i^\dagger +\sum _{i\le j}X_{e,ij}^{eff} a_i^\dagger a_j^\dagger + \sum _{i\le j\le k}X_{e,ijk}^{eff}a_i^\dagger a_j^\dagger a_k^\dagger +\cdots \end{aligned}$$
(20)

The working equation for the excitation energies is,

$$\begin{aligned}{}[H_{eff}^2,X_e^{eff}] = \Delta E_eX_e^{eff}. \end{aligned}$$
(21)

If equations 13, 14 and 15 are not exactly satistfied, \(X_e^{eff}\) can have a constant component as well. However, we ignore it in the present study, since it is expected to be quite small. This statement applies to the conventional coupled cluster equation of motion based approaches as well, since most of them truncate the cluster operator at some low order. The solutions of equation 21 are obtained directly by diagonalization of effective Hamiltonian \(H_{eff}^2\) in the configuration space constructed by the manifold of excitation operator \(X_e^{eff}\). Since the similarity transformations leave the eigen spectrum of an operator unchanged, the eigenvalues of \(H_{eff}^2\) are the same as of the original Hamiltonian. It may be noted that by virtue of equations 13 and 15, the matrix elements of \(H_{eff}^2\) between the vacuum state and the zeroth order excited states are zero. We restrict the excited state space to all states with at most four quanta excitations from all modes.

2.2 Lowest order perturbative approximation to VCCM

As mentioned in the introduction, the solutions of the nonlinear equations in equations 13 and 15 are among major computational bottlenecks for the VCCM. For a four boson approximation of the cluster operator S, the most expensive contribution in equation 13 comes from the quadratic term \([[H,S_4],S_4]\). Since in the bosonic representation the number of vibrational modes (N) plays the role of basis set in the computation of the cluster matrix elements, the calculation of this term scales up to \(N^6\) for each iteration of the solution of equation 13 (considering the proper intermediate storage is done during the computations). Similar situation arises for the determination of the \(\sigma \) matrix elements with equation 15. We use perturbative approximations to reduce the expensive computations of the cluster matrix elements.

There exists a fundamental relation between the coupled-cluster method and perturbation theory. Using this connection, one can construct any finite-order perturbative corrections to energy and effective Hamiltonian through iterations of the coupled cluster equations. To reduce the computational cost of the evaulation of the cluster matrix elements, we approximate both the cluster operators S and \(\sigma \) only up to first order. Since the reference function in our formulation is a variationally optimized Hartree product function, there is no contribution of \(S_1\) operator to the first order. Thus, in the first order approximation of the cluster operator, only \(S_2\), \(S_3\) and \(S_4\) terms will contribute. Essentially, one has to evaluate only the diagonal part of the linear terms \([H,S_2]\), \([H,S_3]\) and \([H,S_4]\). The computational scaling to evaluate these terms are \(N^2\), \(N^3\) and \(N^4\), respectively. Thus, the computational cost to calculate the cluster matrix elements with the present approximation scales as non-iterative \(N^4\), instead of iterative \(N^6\) in the full VCCM with four boson S. Moreover, such approximation also eliminates the effort to compute the cluster matrix elements \(\sigma \). The excitation operator S and de-excitation operator \(\sigma \) are equal up to first order in perturbation.

In the lowest order approximation, the \(H_{eff}^1\) in equation 16 is constructed with only the first order of the cluster operator S,

$$\begin{aligned} H_{eff}^1= H+[H,S^{(1)}]+\frac{1}{2!}[[H,S^{(1)}],S^{(1)}]\nonumber \\&+\frac{1}{3!}[[[H,S^{(1)}],S^{(1)}],S^{(1)}]+\dots \end{aligned}$$
(22)

Following our previous implementation, we consider \(H_{eff}^1\) has at most four boson operator in it. The double similarity transformation effective Hamiltonian \(H_{eff}^2\) is then constructed using only the first order \(\sigma \)

$$\begin{aligned} H_{eff}^2= e^\sigma H_{eff}^1e^{-\sigma }=H_{eff}^1+[\sigma ^{(1)},H_{eff}^1]\nonumber \\&+\frac{1}{2!}[\sigma ^{(1)},[\sigma ^{(1)},H_{eff}^1]]+\cdots \end{aligned}$$

The \(H_{eff}^2\) is also approximated with a maximum of four boson operators.

Thus, the \(H_{eff}^2\) is correct up to first order with respect to both S and \(\sigma \) due to the linear terms of S and \(\sigma \) in the above equation. However, the presence of the nonlinear terms leads to the partial infinite summation of the lower order corrections. We note that the construction of the effective Hamiltonian \(H_{eff}^1\) and \(H_{eff}^2\) scales as \(N^6\).

3 Results and Discussion

We implemented the lowest order perturbative approximation to the VCCM as described above. The vibrational transition energies of water, and 1,1-difluoroethylene (\(1,1\text{-}\mathrm{C}_2\mathrm{F}_2\mathrm{H}_2\)) molecule are calculated using the proposed approximation (VCCM(pt)). The numerical performance with such low order VCCM are analyzed in comparison to the full VCCM results.

3.1 H2O

Our first case study is on water molecule. We compared the transition energies from the lowest order perturbative VCCM method (VCCM(pt)) with full VCCM and converged full VCI results (FVCI) in Table 1. The FVCI calculations were carried out using 8 harmonic oscillator basis (HO) functions for the stretching modes and 16 HO basis functions for the bending mode. The convergence of the VCI transition energies were found with this basis set. The quartic potential energy surface was calculated using Gaussian09 software suit,52 using MP2 method with cc-pVTZ basis sets.

Table 1 Transition energies (in \(\mathrm{cm}^{-1}\)) of \(H_2O\).
Full size table

We find that the VCCM(pt) approximation gives results comparable to the full VCCM. For the fundamentals and two quanta excitations, the deviations between these two methods are usually less than \(3\ \mathrm{cm}^{-1}\). The maximum deviation among these excitations is only about \(5\ \mathrm{cm}^{-1}\) for \(2_2\) state. The VCCM results are close to the FVCI values, except for the transitions \(1_2\) and \(1_13_1\). The difference between the FVCI and full VCCM results for these two transitions are \(11\ \mathrm{cm}^{-1}\) and \(24\ \mathrm{cm}^{-1}\), respectively. Despite of such discrepancy between the VCCM and FVCI results, the perturbative approximation does not bring any substantial changes for the VCCM transition energies of these states. Even for the three quanta excitations, where the differences between the FVCI and converged VCCM values are substantially large, the VCCM(pt) results are close to the converged VCCM results. For instance, the full VCCM transition energy for higher excited state \(1_13_2\) differ by \(307\ \mathrm{cm}^{-1}\) from the FVCI value. The difference between the VCCM(pt) and full VCCM values for this transition is only \(15\ \mathrm{cm}^{-1}\). The standard deviation of VCCM(pt) results from the full VCCM values for all the states reported here is only \(6\ \mathrm{cm}^{-1}\).

We note that the VCCM(pt) transition energies are higher than the full VCCM transition energies for most of the lower lying transitions. Thus, the perturbative approximation overestimates the VCCM for the transition energies. We note that a correlated transition energy can be decomposed in to three components

$$\begin{aligned} \Delta E = \Delta E^0 +\epsilon _{ex} + \epsilon _{g}^{corr} \end{aligned}$$
(23)

Here, \(\Delta E^0\) is the zeroth order transition energy obtained from the EHO approximation, \(\epsilon _{ex}\) is the correlation effect for the excited states, and \(\epsilon _g^{corr}\) is the ground state correlation correction. The excited state correlation energy, \(\epsilon _{ex}\), will have two parts

$$\begin{aligned} \epsilon _{ex} = \epsilon _{mix} + \epsilon _{rel}. \end{aligned}$$
(24)

The \(\epsilon _{mix}\) arises due to mixing among the zeroth order excited states, and \(\epsilon _{rel}\) is the relaxation effect. It can be shown that the \(\epsilon _{mix}\) energy partially cancels out the ground state correlation energy \(\epsilon _g^{corr}\) for the fundamentals. Thus, the transition energy is dominated by the relaxation term \(\epsilon _{rel}\). The over estimation of the VCCM results with the lowest order perturbative approximations indicates that the relaxation effect is underestimated by this approximation.

3.2 1,1-Difluoroethylene

Our second case study is \(1,1\text{-}\mathrm{C}_2\mathrm{F}_2\mathrm{H}_2\) molecule. The quartic PES for this molecule was calculated using B3LYP density functional method along with 6-311G(2d,2p) basis set. We present the VCCM(pt), full VCCM along with the zeroth order Hartree energies for the fundamental transitions in Table 2. The results are compared with the experimental values.53

Table 2 Transition energies (in \(\mathrm{cm}^{-1}\)) of \(\mathrm{F}_2\mathrm{C}_2\mathrm{H}_2\) molecule.
Full size table

Like \(\mathrm{H}_2\mathrm{O}\) molecule, here also, the VCCM(pt) results are comparable with the full VCCM results. For the most of the states, the difference between these two methods are less than \(10\ \mathrm{cm}^{-1}\). The largest difference in the transition energies is found to be only \(20\ \mathrm{cm}^{-1}\) for the fundamental transitions \(3_1\). The standard deviations between these two methods is about \(10\ \mathrm{cm}^{-1}\) for all the fundamental.

Both the VCCM and VCCM(pt) give results close to the experimental results. The standard deviations for the VCCM and VCCM(pt) results with respect to the experimental values are \(20\ \mathrm{cm}^{-1}\) and \(19\ \mathrm{cm}^{-1}\), respectively.

For \(1,1\text{-}\mathrm{C}_2\mathrm{F}_2\mathrm{H}_2\) molecule also, we find that the VCCM(pt) overestimates the transition energy compared to full VCCM, except for the \(3_1\) and \(9_1\) transitions. The relaxation of frequencies causes the changes in the transition energies (other than Fermi coupling). These relaxation energies have the most significant contributions to the transition energies. The VCCM(pt) underestimates these relaxation energies.

Both the \(3_1\) and \(9_1\) states mix strongly with several other states. The transition energies for these strongly coupled states along with their weight of mixing in the VCCM descriptions are presented in Table 3. We find that the perturbative approximation affects the mixing of states significantly. Consequently, the transition energies deviate from the full VCCM. For example, \(3_1\) fundamental mixes strongly with two quanta states \(6_2\), \(11_112_1\), and \(7_18_1\). In the full VCCM description, the fundamental \(3_1\) has 0.49 weight of zeroth order \(3_1\) state and 0.27 weight of zeroth order combination state \(7_18_1\). In the perturbative approximation VCCM(pt), the weight of zeroth order \(3_1\) state decreases to 0.44. Here, the second major contribution comes from the zeroth order \(11_112_1\) state (weight equal to 0.29). The transition energy for \(3_1\) state is found to be \(1357\ \mathrm{cm}^{-1}\) with full VCCM and \(1337\ \mathrm{cm}^{-1}\) with VCCM(pt) approximation. Similarly, in the full VCCM description of the \(6_2\) state, the zeroth order \(3_1\) state has weight 0.25, which becomes only 0.15 in the VCCM(pt) method. The weight of zeroth order \(6_2\) state increases from 0.47 for the full VCCM to 0.61 for the VCCM(pt) approximation for this \(6_2\) transition. We find an increase of \(73\ \mathrm{cm}^{-1}\) in the transition energy for this \(6_2\) transition from the full VCCM to VCCM(pt) method.

Table 3 A comparison of some strongly coupled state of \(\mathrm{F}_2\mathrm{C}_2\mathrm{H}_2\) with VCCM approaches. (Transition energies are in \(\mathrm{cm}^{-1}\)).
Full size table

The perturbative approximation of the VCCM, i.e. VCCM(pt) estimates the state mixing according to the perturbative analysis. Lower the gap in the zeroth order energy levels, greater is the coefficient. Again, we consider the example of mixing of \(6_2\), \(11_112_1\), and \(7_18_1\) states with \(3_1\) fundamental. The zeroth order Hartree energy difference between \(3_1\) state and \(11_112_1\) state is only \(34\ \mathrm{cm}^{-1}\). Such small difference in Hartree energy leads to strong mixing of these two states in the VCCM(pt) description. The \(3_1\) state has 0.29 weight of zeroth order \(11_112_1\) state and the \(11_112_1\) state has 0.20 weight of zeroth order \(3_1\) state. The differences in the Hartree energy of \(7_18_1\) and \(6_2\) states from the Hartree energy of \(3_1\) are 114 and \(168\ \mathrm{cm}^{-1}\), respectively. The weight of zeroth order \(3_1\) state in the \(7_18_1\) and \(6_2\) states are 0.17 and 0.15, respectively.

The transition energies for the two quanta states with the VCCM methods are presented in Table S1 of the supplementary materials. Trend of the transition energies with the VCCM(pt) approximation is similar to the fundamentals for the two quanta excitations, where the mixing of states is not very significant. The VCCM(pt) values are higher than the full VCCM results for almost all the states. Thus, for the two quanta excitations also, the perturbative approximation underestimates the relaxation corrections. However, the deviations between the VCCM and VCCM(pt) transition energies are not very large. In about forty two quanta excited states, the transition energy difference between the VCCM(pt) and full VCCM results are less than \(10\ \mathrm{cm}^{-1}\). The standard deviation between these two methods is about \(14\ \mathrm{cm}^{-1}\) for two quanta states.

4 Conclusions

In this work, we implemented the lowest order perturbative approximations to the VCCM in bosonic representation, by considering only the first order of both the excitation (S) and de-excitation (\(\sigma \)) cluster matrix operators for the construction of the effective Hamiltonian. We made a systematic comparison between the converged VCCM and perturbative approximation with the example of two molecules, \(\hbox {H}_2\mathrm{O}\) and \(1,1\text{-}\mathrm{C}_2\mathrm{F}_2\mathrm{H}_2\). We find that such low order perturbative approximation to the VCCM gives transition energies close the full VCCM method.

The VCCM(pt) usually overestimates the transition energies compared to the full VCCM. The major contribution to the transition energy arises from the relaxation of the excitations, due to the cancellations of excited state correlation and ground state correlation energies. The overestimated transition energies in the perturbative approximation implies that such lowest order perturbation underestimates the relaxation effect. However, such underestimation of the relaxation effects does not affect the accuracy of the transition energies significantly.

The present lowest order perturbative approximation to the cluster operator reduces the computational cost significanly, as the scaling of computation of the S matrix elements reduces from iterative \(N^6\) to non-iterative \(N^4\). The computation of \(\sigma \) matrix elements are avoided by equating them with the S matrix elements, since the S and \(\sigma \) operators are identatical up to first order.

The mixing of states differs significantly with the perturbative approximation compared to the full VCCM. As a results, we get large deviations for the states which coupled strongly with one or more states. The VCCM(pt) transition energies for such strongly coupled states are found to be even smaller than the full VCCM results. The coefficients of mixing of states in the perturbative approximation are determined by the perturbation analysis. Smaller the difference in the zeroth order Hartree energies, greater is the mixing weight.

Overall, we find good accuracy for the transition energies with the VCCM(pt) approach. The results are comparable with the FVCI results or the experimental results. Such approximations will reduce the computational cost to calculate the converged cluster matrix elements significantly without compromising much of the resultant accuracy.