Automatic generation of force fields and property surfaces for use in variational vibrational calculations of anharmonic vibrational energies and zero-point vibrational averaged properties

An automatic and general procedure for the calculation of geometrical derivatives of the energy and general property surfaces for molecular systems is developed and implemented. General expressions for an n-mode representation are derived, where the n-mode representation includes only the couplings between n or less degrees of freedom. The general expressions are specialized to derivative force fields and property surfaces, and a scheme for calculation of the numerical derivatives is implemented. The implementation is interfaced to electronic structure programs and may be used for both ground and excited electronic states. The implementation is done in the context of a vibrational structure program and can be used in combination with vibrational self-consistent field (VSCF), vibrational configuration interaction (VCI), vibrational Moller-Plesset, and vibrational coupled cluster calculations of anharmonic wave functions and calculation of vibrational averaged properties at the VSCF and VCI levels. Sample calculations are presented for fundamental vibrational energies and vibrationally averaged dipole moments and frequency dependent polarizabilities and hyperpolarizabilities of water and formaldehyde.     

Fermi resonance in CO2: a combined electronic coupled-cluster and vibrational configuration-interaction prediction

The authors present a first-principles prediction of the energies of the eight lowest-lying anharmonic vibrational states of CO(2), including the fundamental symmetric stretching mode and the first overtone of the fundamental bending mode, which undergo a strong coupling known as Fermi resonance. They employ coupled-cluster singles, doubles, and (perturbative) triples [CCSD(T) and CCSDT] in conjunction with a range of Gaussian basis sets (up to cc-pV5Z, aug-cc-pVQZ, and aug-cc-pCVTZ) to calculate the potential energy surfaces (PESs) of the molecule, with the errors arising from the finite basis-set sizes eliminated by extrapolation. The resulting vibrational many-body problem is solved by the vibrational self-consistent-field and vibrational configuration-interaction (VCI) methods with the PESs represented by a fourth-order Taylor expansion or by numerical values on a Gauss-Hermite quadrature grid. With the VCI, the best theoretical estimates of the anharmonic energy levels agree excellently with experimental values within 3.5 cm(-1) (the mean absolute deviation). The theoretical (experimental) anharmonic frequencies of the Fermi doublet are 1288.9 (1285.4) and 1389.3 (1388.2) cm(-1).     

Calculation of vibrational infrared intensities and Raman activities using explicit anharmonic wave functions

Methods for automatic computation of IR intensities and Raman activities are described using vibrational self-consistent field (VSCF) and vibrational configuration interaction (VCI) wave functions. Inclusion of effects due to anharmonicity in the potential energy and property surfaces are found to improve the results substantially as compared to experimental data. Sample calculations employing water and formaldehyde are presented, allowing for comparison between different vibrational methods. The convergence with respect to excitation level in VCI and the extent of mode coupling in the potential and property expansions is investigated. In addition, different electronic methods used for generating the potential and property surfaces, namely CCSD, CCSD(T), DFT/B3LYP, and DFT/CAM-B3LYP have been compared. Details of the potential and property surfaces may have significant effects on the IR and Raman intensities.     

Vibrational quasi-degenerate perturbation theory: applications to fermi resonance in CO2, H2CO, and C6H6

A quasi-degenerate perturbation method with vibrational self-consistent field (VSCF) reference wavefunction is developed. It simultaneously accounts for strong anharmonic mode-mode coupling among a few states (static correlation) by a configuration interaction theory and for weak coupling with a vast number of the other states (dynamic correlation) by a perturbation theory. A general formula is derived based on the van Vleck perturbation theory. An algorithm that selects a compact set of the most important VSCF configurations which contribute to the static correlation is proposed and a scheme to limit the number of configurations considered for dynamic correlation is also implemented. This method reproduces the vibrational frequencies of CO2 and H2CO that are subject to the strongest anharmonic mode-mode coupling within 10 cm(-1) of vibrational configuration interaction results in a computational expense reduced by a factor of one to two orders of magnitude. The method also reproduces the infrared absorption of C6H6 in the CH stretching (nu12) frequency region, in which combination tones nu13nu16 and nu2nu13nu18 appear on account of an intensity borrowing from nu12via the anharmonic coupling.     

Calculating molecular vibrational spectra beyond the harmonic approximation

We present a new approach for calculating anharmonic corrections to vibrational frequency calculations. The vibrational wavefunction is modelled using translated Hermite functions thus allowing anharmonic effects to be incorporated directly into the wavefunction whilst still retaining the simplicity of the Hermite basis. We combine this new method with an optimised finite-difference grid for computing the necessary third and fourth nuclear derivatives of the energy. We compare our combined approach to existing anharmonic models—vibrational self-consistent field theory (VSCF), vibrational perturbation theory (VPT), and vibrational configuration interaction theory (VCI)—and find that it is more cost effective than these alternatives. This makes our method well-suited to computing anharmonic corrections for frequencies in medium-sized molecules. Contribution of the Mark S. Gordon 65th Birthday Festschrift Issue.     

A study of the anharmonic effects on the vibrational spectra of a realistic retinal chromophore model

Ab initio and vibrational self-consistent field (VSCF) computations are used to investigate the vibrational normal coordinates of the protonated Schiff base (PSB) 4-cis-gamma,eta-dimethyl-C9H9 NH2+. The ground and the first excited states are investigated. Both harmonic and anharmonic frequencies for the first three overtones of the ground and first excited states are reported. Special attention is payed to the discussion of the normal coordinates modes that involve the central C=C bond which plays a significant role in the isomerization process.     

Multi-level vibrational SCF calculations and FTIR measurements on furazan

The structure and infrared spectrum of furazan (1,2,5-oxadiazole) were studied by vibrational SCF (VSCF) and configuration interaction (VCI) calculations based on a high quality potential derived from electronic structure calculations up to the CCSD(T)/aug-cc-pCVQZ level. In addition gas-phase FTIR measurements were performed, which allowed for several corrections in the spectrum of the first vibrational overtones. Excellent agreement was found between the computed and the experimental results. Dedicated to Professor Hermann Stoll on occasion of his 60th birthday.     

Accurate ab initio determination of spectroscopic and thermochemical properties of mono- and dichlorocarbenes

The best technically feasible values for the triplet-singlet energy gap and the enthalpies of formation of the HCCl and CCl2 radicals have been determined through the focal-point approach. The electronic structure computations were based on high-level coupled cluster (CC) methods, up to quadruple excitations (CCSDTQ), and large-size correlation-consistent basis sets, ranging from aug-cc-pVDZ to aug-cc-pV6Z, followed by extrapolation to the complete basis set limit. Small corrections due to core correlation, relativistic effects, diagonal Born-Oppenheimer correction, as well as harmonic and anharmonic zero-point vibrational energy corrections have been taken into account. The final estimates for the triplet-singlet energy gap, T0(?), are 2170+/-40 cm-1 for HCCl and 7045+/-60 cm-1 for CCl2, favoring the singlet states in both cases. Complete quartic force fields in internal coordinates have been computed for both the X and ? states of both radicals at the frozen-core CCSD(T)/aug-cc-pVQZ level. Using these force fields vibrational energy levels of {HCCl, DCCl, CCl2} up to {6000, 5000, 7000} cm-1 were calculated both by second-order vibrational perturbation theory (VPT2) and variationally. These results, especially the variational ones, show excellent agreement with the experimentally determined energy levels. The enthalpies of formation of HCCl (X1A') and CCl2(X1A1), at 0 K, are 76.28+/-0.20 and 54.54+/-0.20 kcal mol-1, respectively.     

Complexes of type C6H7(+)·L (L = N2 and CO2) studied by explicitly correlated coupled cluster theory

Complexes of the benzenium ion (C(6)H(7)(+)) with N(2) or CO(2) have been studied by explicitly correlated coupled cluster theory at the CCSD(T)-F12x (x = a, b) level [T. B. Adler et al., J. Chem. Phys. 127, 221106 (2007)] and the double-hybrid density functional B2PLYP-D [T. Schwabe and S. Grimme, Phys. Chem. Chem. Phys. 9, 3397 (2007)]. Improved harmonic vibrational wavenumbers for C(6)H(7)(+) have been obtained by CCSD(T?)-F12a calculations with the VTZ-F12 basis set. Combining them with previous B2PLYP-D anharmonic contributions we arrive at anharmonic wavenumbers which are in excellent agreement with recent experimental data from p-H(2) matrix isolation IR spectroscopy [M. Bahou et al., J. Chem. Phys. 136, 154304 (2012)]. The energetically most favourable conformer of C(6)H(7)(+)·N(2) shows a π-bonded structure similar to C(6)H(7)(+)·Rg (Rg = Ne, Ar) [P. Botschwina and R. Oswald, J. Phys. Chem. A 115, 13664 (2011)] with D(e) ≈ 870 cm(-1). For C(6)H(7)(+)·CO(2), a slightly lower energy is calculated for a conformer with the CO(2) ligand lying in the ring-plane of the C(6)H(7)(+) moiety (D(e) ≈ 1508 cm(-1)). It may be discriminated from other conformers through a strong band predicted at 1218 cm(-1), red-shifted by 21 cm(-1) from the corresponding band of free C(6)H(7)(+).     

Anharmonic rovibrational analysis for disilacyclopropenylidene (Si2CH2)

The global minimum on the Si(2)CH(2) electronic singlet potential energy surface has been theoretically predicted to be a peculiar hydrogen bridged (Si···H···Si) disilacyclopropenylidene structure (Si(2)CH(2)). An accurate quartic force field for Si(2)CH(2) has been determined employing ab initio coupled-cluster theory with single and double excitations and a perturbative treatment for triple excitations [CCSD(T)], in combination with the correlation consistent core-valence quadruple zeta (cc-pCVQZ) basis set. The vibration-rotation coupling constants, equilibrium and zero-point vibration corrected rotational constants, centrifugal distortion constants, and harmonic and fundamental vibrational frequencies for six isotopologues of Si(2)CH(2) are predicted using vibrational second-order perturbation theory (VPT2). The anharmonic corrections for the vibrational motions involving the H bridged bonds are found to be more than 5% with respect to the corresponding harmonic vibrational frequencies. In this light, an experimental detection and characterization of disilacyclopropenylidene (Si(2)CH(2)) is highly desired.     

Size-extensive vibrational self-consistent field method

The vibrational self-consistent field (VSCF) method is a mean-field approach to solve the vibrational Schro?dinger equation and serves as a basis of vibrational perturbation and coupled-cluster methods. Together they account for anharmonic effects on vibrational transition frequencies and vibrationally averaged properties. This article reports the definition, programmable equations, and corresponding initial implementation of a diagrammatically size-extensive modification of VSCF, from which numerous terms with nonphysical size dependence in the original VSCF equations have been eliminated. When combined with a quartic force field (QFF), this compact and strictly size-extensive VSCF (XVSCF) method requires only quartic force constants of the ?(4)V/?Q(i)(2)?Q(j)(2) type, where V is the electronic energy and Q(i) is the ith normal coordinate. Consequently, the cost of a XVSCF calculation with a QFF increases only quadratically with the number of modes, while that of a VSCF calculation grows quartically. The effective (mean-field) potential of XVSCF felt by each mode is shown to be harmonic, making the XVSCF equations subject to a self-consistent analytical solution without matrix diagonalization or a basis-set expansion, which are necessary in VSCF. Even when the same set of force constants is used, XVSCF is nearly three orders of magnitude faster than VSCF implemented similarly. Yet, the results of XVSCF and VSCF are shown to approach each other as the molecular size is increased, implicating the inclusion of unnecessary, nonphysical terms in VSCF. The diagrams of the XVSCF energy expression and their evaluation rules are also proposed, underscoring their connected structures.     

Highly accurate determination of the electron affinity of SF6 and analysis of structure and photodetachment spectrum of SF6-

The title system is thoroughly investigated by high-level electronic structure techniques and nuclear quantum dynamics calculations. Equilibrium geometries and harmonic frequencies are determined by coupled-cluster singles doubles [CCSD(T)] calculations with large AO basis sets. A C(4v) distorted geometry is found for the anion in contrast to previous assumptions. This is explained by the bonding situation in the electronic ground state and possible vibronic interactions with higher electronic states. The computed adiabatic electron affinity of 0.73 eV is considerably lower than the currently recommended value. Analysis of the electronic states of the anion shows that the σ* ground state at equilibrium position corresponds to a highly excited state at the neutral's geometry where the ground state is either a very weakly bound or scattering state. If the electron is captured by this latter state, a nonadiabatic transition to the σ* state followed by internal vibrational redistribution could explain the formation of a stable anion. The C(4v) distortion of the equilibrium geometry is essential for the explanation of recently measured photodetachment spectra. Since the distortion leads to six equivalent minima with very low barriers, an anharmonic potential energy surface (PES) of the four relevant vibrational modes is constructed and fitted to CCSD(T) computed energies. The remaining 11 modes are treated as harmonic oscillators. The vibrational dynamics of the anion is studied by diagonalization of the Hamiltonian in the basis of the neutral's eigenstates. The computed photoelectron spectra are in good agreement with recent experiments and demonstrate the quality of the PES and that C(4v) distortion is responsible for the observed irregularities. However, thermal effects play a significant role for the shape of the spectra because many low-lying initial states are populated.     

Vibrational contributions to indirect spin-spin coupling constants calculated via variational anharmonic approaches

Zero-point vibrational contributions to indirect spin-spin coupling constants for N2, CO, HF, H2O, C2H2, and CH4 are calculated via explicitly anharmonic approaches. Thermal averages of indirect spin-spin coupling constants are calculated for the same set of molecules and for C2X4, X = H, F, Cl. Potential energy surfaces have been calculated on a grid of points and analytic representations have been obtained by a linear least-squares fit in a direct product polynomial basis. Property surfaces have been represented by a fourth-order Taylor expansion around the equilibrium geometry. The electronic structure calculations employ density functional theory, and vibrational contributions to indirect spin-spin coupling constants are calculated employing vibrational self-consistent-field and vibrational configuration-interaction methods. The performance of vibrational perturbation theory and various approximate variational calculations are discussed. Thermal averages are computed by state-specific and virtual vibrational self-consistent-field methods.     

Vibrational spectroscopy of protonated imidazole and its complexes with water molecules: ab initio anharmonic calculations and experiments

The results of anharmonic frequency calculations on neutral imidazole (C3N2H4, Im), protonated imidazole (ImH+), and its complexes with water (ImH+)(H2O)n, are presented and compared to gas phase infrared photodissociation spectroscopy (IRPD) data. Anharmonic frequencies are obtained via ab initio vibrational self-consistent field (VSCF) calculations taking into account pairwise interactions between the normal modes. The key results are: (1) Prediction of anharmonic vibrational frequencies on an MP2 ab initio potential energy surface show excellent agreement with experiment and outstanding improvement over the harmonic frequencies. For example, the ab initio calculated anharmonic frequency for (ImH+)(H2O)N2 exhibits an overall average percentage error of 0.6% from experiment. (2) Anharmonic vibrational frequencies calculated on a semiempirical potential energy surface fitted to ab initio harmonic data represents spectroscopy well, particularly for water complexes. As an example, anharmonic frequencies for (ImH+)H2O and (ImH+)(H2O)2 show an overall average deviation of 1.02% and 1.05% from experiment, respectively. This agreement between theory and experiment also supports the validity and use of the pairwise approximation used in the calculations. (3) Anharmonic coupling due to hydration effects is found to significantly reduce the vibrational frequencies for the NH stretch modes. The frequency of the NH stretch is observed to increase with the removal of a water molecule or replacement of water with N2. This result also indicates the ability of the VSCF method to predict accurate frequencies in a matrix environment. The calculation provides insights into the nature of anharmonic effects in the potential surface. Analysis of percentage anharmoncity in neutral Im and ImH+ shows a higher percentage anharmonicity in the NH and CH stretch modes of neutral Im. Also, we observe that anharmonicity in the NH stretch modes of ImH+ have some contribution from coupling effects, while that of neutral Im has no contribution whatsoever from mode-mode coupling. It is concluded that the incorporation of anharmonic effects in the calculation brings theory and experiment into much closer agreement for these systems.     

The impact of approximate VSCF schemes and curvilinear coordinates on the anharmonic vibrational frequencies of formamide and thioformamide

The accuracy of the vibrational self-consistent field (VSCF) method for the computation of anharmonic vibrational frequencies in the infrared (IR) spectrum of formamide and thioformamide is investigated. The importance of triple potentials in the commonly used hierarchical expansion of the potential energy surface (PES) is studied in detail. The PES is expanded in terms of Cartesian as well as internal coordinate normal mode displacements. It is found that triples play an important role when using rectilinear coordinates. A VSCF computation based on rectilinear displacements exhibits serious shortcomings which are only remedied by a large vibrational configuration interaction (VCI) treatment including triple potentials. These limitations are partially removed when using curvilinear coordinates. The merits and disadvantages of either type of displacements for the generation of the PES are discussed.     

Franck-Condon simulation of the single vibronic level emission spectra of HSiF and DSiF including anharmonicity

Potential energy functions (PEFs) of the X (1)A(') and A (1)A(") states of HSiF have been computed using the coupled-cluster single-double plus perturbative triple excitations and complete-active-space self-consistent-field multireference internally contracted configuration interaction methods, respectively, employing augmented correlation-consistent polarized-valence quadruple-zeta basis sets. For both electronic states of HSiF and DSiF, anharmonic vibrational wavefunctions and energies of all three modes have been calculated variationally with the ab initio PEFs and using Watson's Hamiltonian for nonlinear molecules. Franck-Condon factors between the two electronic states, allowing for Duschinsky rotation, were computed using the calculated anharmonic vibrational wavefunctions. These Franck-Condon factors were used to simulate the single vibronic level (SVL) emission spectra recently reported by Hostutler et al. in J. Chem. Phys. 114, 10728 (2001). Excellent agreement between the simulated and observed spectra was obtained for the A (1)A(")(1,0,0)-->X (1)A(') SVL emission of HSiF. Discrepancies between the simulated and observed spectra of the A (1)A(")(0,1,0) and (1,1,0) SVL emissions of HSiF have been found. These are most likely, partly due to experimental deficiencies and, partly to inadequacies in the ab initio levels of theory employed in the calculation of the PEFs. Based on the computed Franck-Condon factors, minor revisions of previous vibrational assignments are suggested. The calculated anharmonic wave functions of higher vibrational levels of the X (1)A(') state show strong mixings between the three vibrational modes of HSi stretching, bending, and SiF stretching.     

Exploring the effect of anharmonicity of molecular vibrations on thermodynamic properties

Thermodynamic properties of selected small and medium size molecules were calculated using harmonic and anharmonic vibrational frequencies. Harmonic vibrational frequencies were obtained by normal mode analysis, whereas anharmonic ones were calculated using the vibrational self-consistent field (VSCF) method. The calculated and available experimental thermodynamic data for zero point energy, enthalpy, entropy, and heat capacity are compared. It is found that the anharmonicity and coupling of molecular vibrations can play a significant role in predicting accurate thermodynamic quantities. Limitations of the current VSCF method for low frequency modes have been partially removed by following normal mode displacements in internal, rather than Cartesian, coordinates.     

Anharmonic overtone and combination states of glycine and two model peptides examined by vibrational self-consistent field theory

In this paper, the application of the vibrational self-consistent field (VSCF) and correction-corrected VSCF methods for calculating anharmonic parameters, including transition frequency, transition intensity and dipole, and vibrational anharmonicity of 3N-6 normal modes for formamide, glycine, N-methylacetamide and their deuterated derivatives are explored mainly at the level of density functional theory. The computed fundamental anharmonic frequencies are found to be in reasonable agreement with experimental results. Diagonal anharmonicities of the second overtone states were examined for multiple normal modes, whose magnitudes were found to correlate well with those of the first overtone states in the three small molecules. The results show that the VSCF-based approach can be utilized to predict anharmonic parameters of higher vibrational states that are essential to understanding multi-pulse infrared nonlinear experiments of peptides.     

Size-extensive vibrational self-consistent field methods with anharmonic geometry corrections

In the size-extensive vibrational self-consistent field (XVSCF) method introduced earlier [M. Ke?eli and S. Hirata, J. Chem. Phys. 135, 134108 (2011)], only a small subset of even-order force constants that can form connected diagrams were used to compute extensive total energies and intensive transition frequencies. The mean-field potentials of XVSCF formed with these force constants have been shown to be effectively harmonic, making basis functions, quadrature, or matrix diagonalization in the conventional VSCF method unnecessary. We introduce two size-consistent VSCF methods, XVSCF(n) and XVSCF[n], for vibrationally averaged geometries in addition to energies and frequencies including anharmonic effects caused by up to the nth-order force constants. The methods are based on our observations that a small number of odd-order force constants of certain types can form open, connected diagrams isomorphic to the diagram of the mean-field potential gradients and that these nonzero gradients shift the potential minima by intensive amounts, which are interpreted as anharmonic geometry corrections. XVSCF(n) evaluates these mean-field gradients and force constants at the equilibrium geometry and estimates this shift accurately, but approximately, neglecting the coupling between these two quantities. XVSCF[n] solves the coupled equations for geometry corrections and frequencies with an iterative algorithm, giving results that should be identical to those of VSCF when applied to an infinite system. We present the diagrammatic and algebraic definitions, algorithms, and initial implementations as well as numerical results of these two methods. The results show that XVSCF(n) and XVSCF[n] reproduce the vibrationally averaged geometries of VSCF for naphthalene and anthracene in their ground and excited vibrational states accurately at fractions of the computational cost.