Vibrational Coupled Cluster Theory Research Articles

Efficient time-dependent vibrational coupled cluster computations with time-dependent basis sets at the two-mode coupling level: Full and hybrid TDMVCC[2

The computation of the nuclear quantum dynamics of molecules is challenging, requiring both accuracy and efficiency to be applicable to systems of interest. Recently, theories have been developed for employing time-dependent basis functions (denoted modals) with vibrational coupled cluster theory (TDMVCC). The TDMVCC method was introduced along with a pilot implementation, which illustrated good accuracy in benchmark computations. In this paper, we report an efficient implementation of TDMVCC, covering the case where the wave function and Hamiltonian contain up to two-mode couplings. After a careful regrouping of terms, the wave function can be propagated with a cubic computational scaling with respect to the number of degrees of freedom. We discuss the use of a restricted set of active one-mode basis functions for each mode, as well as two interesting limits: (i) the use of a full active basis where the variational modal determination amounts essentially to the variational determination of a time-dependent reference state for the cluster expansion; and (ii) the use of a single function as an active basis for some degrees of freedom. The latter case defines a hybrid TDMVCC/TDH (time-dependent Hartree) approach that can obtain even lower computational scaling. The resulting computational scaling for hybrid and full TDMVCC[2] is illustrated for polyaromatic hydrocarbons with up to 264 modes. Finally, computations on the internal vibrational redistribution of benzoic acid (39 modes) are used to show the faster convergence of TDMVCC/TDH hybrid computations towards TDMVCC compared to simple neglect of some degrees of freedom.

A general implementation of time-dependent vibrational coupled-cluster theory.

The first general excitation level implementation of the time-dependent vibrational coupled cluster (TDVCC) method introduced in a recent publication [J. Chem. Phys. 151, 154116 (2019)] is presented. The general framework developed for time-independent vibrational coupled cluster (VCC) calculations has been extended to the time-dependent context. This results in an efficient implementation of TDVCC with general coupling levels in the cluster operator and Hamiltonian. Thus, the convergence of the TDVCC[k] hierarchy toward the complete-space limit can be studied for any sum-of-product Hamiltonian. Furthermore, a scheme for including selected higher-order excitations for a subset of modes is introduced and studied numerically. Three different definitions of the TDVCC autocorrelation function (ACF) are introduced and analyzed in both theory and numerical experiments. Example calculations are presented for an array of systems including imidazole, formyl fluoride, formaldehyde, and a reduced-dimensionality bithiophene model. The results show that the TDVCC[k] hierarchy converges systematically toward the full-TDVCC limit and that the implementation allows accurate quantum-dynamics simulations of large systems to be performed. Specifically, the intramolecular vibrational-energy redistribution of the 21-dimensional imidazole molecule is studied in terms of the decay of the ACF. Furthermore, the importance of product separability in the definition of the ACF is highlighted when studying non-interacting subsystems.

Vibrational Coupled Cluster Computations in Polyspherical Coordinates with the Exact Analytical Kinetic Energy Operator.

We present the first use of curvilinear vibrational coordinates, specifically polyspherical coordinates, in combination with vibrational coupled cluster theory. The polyspherical coordinates are used in the context of both the adaptive density-guided approach to potential energy surface construction and in the subsequent vibrational coupled cluster calculations of anharmonic vibrational states. Results obtained based on the polyspherical coordinate parametrization are compared to results obtained with the use of rectilinear vibrational coordinates, namely, normal coordinates and hybrid optimized and localized coordinates for the formaldehyde molecule. This comparison is carried out with the full vibrational configuration interaction model, using the respective fully coupled potential energy surfaces and untruncated kinetic energy operators. The polyspherical coordinates are shown to facilitate an acceleration of convergence for truncated methods when compared to the use of normal coordinates. We furthermore report on calculations on the hydrogen peroxide molecule in the polyspherical coordinate parametrization. The polyspherical vibrational coordinates are shown to perform very well, even for truncated methods, especially when considering the difficulty that rectilinear vibrational coordinates can exhibit in treating complicated internal molecular motion.

Time-dependent vibrational coupled cluster theory: Theory and implementation at the two-mode coupling level.

Equations are derived for the time evolution of time-dependent vibrational coupled cluster (TDVCC) wave functions covering both the TDVCC ket state and the associated so-called Λ bra state. The equations are implemented in the special case of both the Hamiltonian and the cluster operator containing at most two-mode coupling terms. The nontrivial behavior of the evolution of norm, energy, and expectation values due to the nonunitary time-evolution of the nonvariational TDVCC theory is analyzed theoretically and confirmed in numerical experiments that also include time-dependent Hamiltonians. In the spirit of time-independent size-consistency analysis, the separability of both the coupled cluster and Λ states for noninteracting systems is studied. While the coupled cluster state clearly has the correct behavior, the behavior of the Λ state is more intricate, and the consequence for different properties is shown theoretically and numerically. Overall, the numerical experiments show that TDVCC in incomplete expansions gives higher accuracy than a standard linear variational wave function parameterization with the same number of independent parameters, while equivalent results are obtained for complete expansions. The efficiency of the methodology is illustrated in computations on polycyclic aromatic hydrocarbons with up to 156 modes.

Similarity-transformed equation-of-motion vibrational coupled-cluster theory.

A similarity-transformed equation-of-motion vibrational coupled-cluster (STEOM-XVCC) method is introduced as a one-mode theory with an effective vibrational Hamiltonian, which is similarity transformed twice so that its lower-order operators are dressed with higher-order anharmonic effects. The first transformation uses an exponential excitation operator, defining the equation-of-motion vibrational coupled-cluster (EOM-XVCC) method, and the second uses an exponential excitation-deexcitation operator. From diagonalization of this doubly similarity-transformed Hamiltonian in the small one-mode excitation space, the method simultaneously computes accurate anharmonic vibrational frequencies of all fundamentals, which have unique significance in vibrational analyses. We establish a diagrammatic method of deriving the working equations of STEOM-XVCC and prove their connectedness and thus size-consistency as well as the exact equality of its frequencies with the corresponding roots of EOM-XVCC. We furthermore elucidate the similarities and differences between electronic and vibrational STEOM methods and between STEOM-XVCC and vibrational many-body Green's function theory based on the Dyson equation, which is also an anharmonic one-mode theory. The latter comparison inspires three approximate STEOM-XVCC methods utilizing the common approximations made in the Dyson equation: the diagonal approximation, a perturbative expansion of the Dyson self-energy, and the frequency-independent approximation. The STEOM-XVCC method including up to the simultaneous four-mode excitation operator in a quartic force field and its three approximate variants are formulated and implemented in computer codes with the aid of computer algebra, and they are applied to small test cases with varied degrees of anharmonicity.

Tensor-decomposed vibrational coupled-cluster theory: Enabling large-scale, highly accurate vibrational-structure calculations.

A new implementation of vibrational coupled-cluster (VCC) theory is presented, where all amplitude tensors are represented in the canonical polyadic (CP) format. The CP-VCC algorithm solves the non-linear VCC equations without ever constructing the amplitudes or error vectors in full dimension but still formally includes the full parameter space of the VCC[n] model in question resulting in the same vibrational energies as the conventional method. In a previous publication, we have described the non-linear-equation solver for CP-VCC calculations. In this work, we discuss the general algorithm for evaluating VCC error vectors in CP format including the rank-reduction methods used during the summation of the many terms in the VCC amplitude equations. Benchmark calculations for studying the computational scaling and memory usage of the CP-VCC algorithm are performed on a set of molecules including thiadiazole and an array of polycyclic aromatic hydrocarbons. The results show that the reduced scaling and memory requirements of the CP-VCC algorithm allows for performing high-order VCC calculations on systems with up to 66 vibrational modes (anthracene), which indeed are not possible using the conventional VCC method. This paves the way for obtaining highly accurate vibrational spectra and properties of larger molecules.

On the choice electronic structure method to calculate the quartic potential energy surface for the vibrational calculation of polyatomic molecules

A systematic study is made on the effects of electronic structure calculations on anharmonic spectra of polyatomic molecules. Our study is focused on the choice of electronic structure method and basis set to calculate the quartic potential energy surface (PES). We used two correlated methods Moller–Plesset perturbation theory and density functional theory with B3LYP functional and two different types of basis sets, aug-cc-pVTZ and 6-311G(2d,2p) to calculate the PES and linear DMS. For the vibrational description, we used the vibrational self-consistent field theory and the vibrational coupled cluster theory in bosonic representation.

Higher-order diagrammatic vibrational coupled-cluster theory.

Diagrammatically size-consistent and basis-set-free vibrational coupled-cluster (XVCC) theory for both zero-point energies and transition frequencies of a molecule, the latter through the equation-of-motion (EOM) formalism, is defined for an nth-order Taylor-series potential energy surface (PES). Quantum-field-theoretical tools (the rules of normal-ordered second quantization and Feynman-Goldstone diagrams) for deriving their working equations are established. The equations of XVCC and EOM-XVCC including up to the mth-order excitation operators are derived and implemented with the aid of computer algebra in the range of 1 ≤ m ≤ 8. Algorithm optimizations known as strength reduction, intermediate reuse, and factorization are carried out before code generation, reducing the cost scaling of the mth-order XVCC and EOM-XVCC in an nth-order Taylor-series PES (m ≥ n) to the optimal value of O(N(m+⌊n/2⌋)), where N is the number of modes. The calculated zero-point energies and frequencies of fundamentals, overtones, and combinations as well as Fermi-resonant modes display rapid and nearly monotonic convergence with m towards the exact values for the PES. The theory with the same excitation rank as the truncation order of the Taylor-series PES (m = n) seems to strike the best cost-accuracy balance, achieving the accuracy of a few tenths of cm(-1) for transitions involving (m - 3) modes and of a few cm(-1) for those involving (m - 2) modes. The relationships between XVCC and the vibrational coupled-cluster theories of Prasad and coworkers and of Christiansen and coworkers as well as the size-extensive vibrational self-consistent-field and many-body perturbation theories are also elucidated.

Approximate inclusion of four-mode couplings in vibrational coupled-cluster theory

The vibrational coupled cluster (VCC) equations are analyzed in terms of vibrational Mo̸ller-Plesset perturbation theory aiming specifically at the importance of four-mode couplings. Based on this analysis, new VCC methods are derived for the calculation of anharmonic vibrational energies and vibrational spectra using vibrational coupled cluster response theory. It is shown how the effect of four-mode coupling and excitations can be efficiently and accurately described using approximations for their inclusion. Two closely related approaches are suggested. The computational scaling of the so-called VCC[3pt4F] method is not higher than the fifth power in the number of vibrational degrees of freedom when up to four-mode coupling terms are present in the Hamiltonian and only fourth order when only up to three-mode couplings are present. With a further approximation, one obtains the VCC[3pt4] model which is shown to scale with at most the fourth power in the number of vibrational degrees of freedom for Hamiltonians with both three- and four-mode coupling levels, while sharing the most important characteristics with VCC[3pt4F]. Sample calculations reported for selected tetra-atomic molecules as well as the larger dioxirane and ethylene oxide molecules support that the new models are accurate and useful.

Automatic derivation and evaluation of vibrational coupled cluster theory equations

A scheme for automatic derivation and evaluation of the expressions occurring in vibrational coupled cluster theory is introduced. The method is based on a Baker-Campbell-Hausdorff expansion of the similarity transformed Hamiltonian and is general both with respect to the excitation level in the parameter space and the mode coupling level in the Hamiltonian. In addition to deriving general expressions, intermediates that lower the computational scaling are automatically detected. The final equations are then evaluated. Due to the commutator based nature of the algorithm, it is also applicable to the evaluation of quantities needed for response theory. Different aspects of the theory and implementation are illustrated by calculations on model systems. Furthermore, all fundamental excitation energies of ethylene oxide are calculated.

Vibrational coupled cluster theory with full two-mode and approximate three-mode couplings: The VCC[2pt3] model

Vibrational coupled cluster (VCC) calculations of molecular vibrational energy levels can be characterized by the number of modes coupled in the Hamiltonian operator and the number of modes simultaneously excited in the parameter space. We propose a VCC model which includes all two-mode couplings in the Hamiltonian and excitation space but only an approximate treatment of three-mode couplings. The approximation is based on a perturbational analysis and the introduced concepts can also be used for even more accurate treatments. The method is iterative and allows the use of VCC response theory to obtain excitation energies. Furthermore, the method is shown to scale with the number of vibrational modes to the third power which is no higher than the corresponding VCC model with only two-mode couplings. Encouraging benchmark calculations are given for a test set of three- and four-atomic molecules. The fundamentals of the larger ethylene oxide molecule have been calculated as well using a grid-based potential energy surface obtained from electronic coupled cluster theory with singles, doubles, and perturbative triples (CCSD(T)).

Towards fast computations of correlated vibrational wave functions: Vibrational coupled cluster response excitation energies at the two-mode coupling level

An efficient implementation of vibrational coupled cluster theory with two-mode excitations and a two-mode Hamiltonian is described. The algorithm is shown to scale cubically with respect to the number of modes which is identical to the scaling of the corresponding vibrational configuration interaction algorithm. This is achieved through the use of special intermediates. The same algorithm can also be used in vibrational Møller-Plesset calculations. To improve performance, screening techniques have been implemented as well. Test calculations on polyaromatic hydrocarbons with up to 264 coupled modes and model systems with up to 1140 modes are used to illustrate the various features of the algorithm.

Coupled Cluster Theory with Emphasis on Selected New Developments

Coupled cluster (CC) methods for the description of the correlated motion of electrons and nuclei are reviewed with emphasis on selected new initiatives. The basic aspects of standard electronic CC theory are described including the rationale behind the most widely used methods like coupled cluster singles and doubles (CCSD) and the CCSD(T) approach. The hierarchy of coupled cluster models consisting of coupled cluster singles (CCS), CC2, CCSD and CC3 is also described. A brief account of the theory behind the calculation of molecular properties using CC methods, and the description of response properties and excited states using CC response theory is followed by a discussion of the use of CC theory in the context of effective models for describing molecules in solution. In another part of the review we consider recent initiatives aimed at the development of coupled cluster methods for describing the correlated motion of the atomic nuclei. A recently developed second quantization formulation of many-mode dynamics for distinguishable degrees of freedom forms the basis for developing new quantum dynamical methods in particular vibrational coupled cluster (VCC) methods. The VCC theory is reviewed and discussed in comparison with vibrational configuration interaction (VCI), and vibrational Moller–Plesset (VMP) perturbation theory. The review concludes with a discussion of some important future research topics

Vibrational coupled cluster theory.

The theory and first implementation of a vibrational coupled cluster (VCC) method for calculations of the vibrational structure of molecules is presented. Different methods for introducing approximate VCC methods are discussed including truncation according to a maximum number of simultaneous mode excitations as well as an interaction space order concept is introduced. The theory is tested on calculation of anharmonic frequencies for a three-mode model system and a formaldehyde quartic force field. The VCC method is compared to vibrational self-consistent-field, vibrational Møller-Plesset perturbation theory, and vibrational configuration interaction (VCI). A VCC calculation typically gives higher accuracy than a corresponding VCI calculation with the same number of parameters and the same formal operation count.