Direct calculation of anharmonic vibrational states of polyatomic molecules using potential energy surfaces calculated from density functional theory

On the choice of coordinates in anharmonic theoretical vibrational spectroscopy: Harmonic vs. anharmonic coupling in vibrational configuration interaction

By a suitable choice of coordinates, the computational effort required for calculations of anharmonic vibrational spectra can be reduced significantly. By using suitable localized-mode coordinates obtained from an orthogonal transformation of the conventionally used normal-mode coordinates, anharmonic couplings between modes can be significantly reduced. However, such a transformation introduces harmonic couplings between the localized modes. To elucidate the role of these harmonic couplings, we consider the vibrational self-consistent field (VSCF)/vibrational configuration interaction (VCI) calculations for both few-mode model systems and for ethene as a molecular test case. We show that large harmonic couplings can result in significant errors in localized-mode L-VSCF/L-VCI calculations and study the convergence with respect to the size of the VCI excitation space. To further elucidate the errors introduced by harmonic couplings, we discuss the connection between L-VSCF/L-VCI and vibrational exciton models. With the help of our results, we propose an algorithm for the localization of normal modes in suitable subsets that are chosen to strictly limit the errors introduced by the harmonic couplings while still leading to maximally localized modes.

Vibrational anharmonicity of small gold and silver clusters using the VSCF method

We study the vibrational spectra of small neutral gold (Au₂–Au₁₀) and silver (Ag₂–Au₅) clusters using the vibrational self-consistent field method (VSCF) in order to account for anharmonicity.

Generalized Vibrational Perturbation Theory for Rotovibrational Energies of Linear, Symmetric and Asymmetric Tops: Theory, Approximations, and Automated Approaches to Deal with Medium-to-Large Molecular Systems

Models going beyond the rigid-rotor and the harmonic oscillator levels are mandatory for providing accurate theoretical predictions for several spectroscopic properties. Different strategies have been devised for this purpose. Among them, the treatment by perturbation theory of the molecular Hamiltonian after its expansion in power series of products of vibrational and rotational operators, also referred to as vibrational perturbation theory (VPT), is particularly appealing for its computational efficiency to treat medium-to-large systems. Moreover, generalized (GVPT) strategies combining the use of perturbative and variational formalisms can be adopted to further improve the accuracy of the results, with the first approach used for weakly coupled terms, and the second one to handle tightly coupled ones. In this context, the GVPT formulation for asymmetric, symmetric, and linear tops is revisited and fully generalized to both minima and first-order saddle points of the molecular potential energy surface. The computational strategies and approximations that can be adopted in dealing with GVPT computations are pointed out, with a particular attention devoted to the treatment of symmetry and degeneracies. A number of tests and applications are discussed, to show the possibilities of the developments, as regards both the variety of treatable systems and eligible methods.

New developments of a multifrequency virtual spectrometer: stereo-electronic, dynamical, and environmental effects on chiroptical spectra

Computational spectroscopy has recently evolved from a field reserved for specialists toward a general tool allowing interpretations and analyses of experimental results. However, the current practice of providing tables of transitions for rigid geometries, possibly tuned by phenomenological broadening, is by far too naive. In order to improve this situation in the last few years we have been developing a general, robust, and user-friendly virtual spectrometer (VS) able to complement experimental studies for complex systems in condensed phases. The VS is based on flexible graphical pre- and postprocessing tools interfaced with general number-crunching software. This last tool is rooted in several electronic structure methodologies (DFT, TD-DFT, post-Hartree-Fock), powerful discrete/continuum models for describing environmental effects, and general vibrational and vibronic models. These last topics are the main focus of this work, which sketches our latest developments related to effective inclusion of anharmonic contributions, together with time-independent and/or time-dependent descriptions of vibronic transitions including Franck-Condon, Herzberg-Teller, and Duschinsky effects. Some test cases are described in some detail with the aim of showing the role of different effects in ruling vibrational (VCD) and electronic (ECD, CPL) chiral spectroscopies.

Efficient calculation of anharmonic vibrational spectra of large molecules with localized modes

The analysis and interpretation of the vibrational spectra of complex (bio)molecular systems, such as polypeptides and proteins, requires support from quantum-chemical calculations. Such calculations are currently restricted to the harmonic approximation. Here, we show how one of the main bottlenecks in such calculations, the evaluation of the potential energy surface, can be overcome by using localized modes instead of the commonly employed normal modes. We apply such local vibrational self-consistent field (L-VSCF) and vibrational configuration interaction (L-VCI) calculations to a cyclic water tetramer and a helical hexa-alanine peptide. The results show that the use of localized modes is equivalent to the commonly used normal modes, but offers several advantages. First, a faster convergence with respect to the excitation level is observed in L-VCI calculations. Second, the localized modes provide a reduced representation of the couplings between modes that show a regular coupling pattern. This can be used to disregard a significant number of small two-mode potentials a priori. Several such reduced coupling approximations are explored, and we show that the number of single-point calculations required to evaluate the potential energy surface can be significantly reduced without introducing noticeable errors in the resulting vibrational spectra.

Optimized Structure and Vibrational Properties by Error Affected Potential Energy Surfaces

The precise theoretical determination of the geometrical parameters of molecules at the minima of their potential energy surface and of the corresponding vibrational properties are of fundamental importance for the interpretation of vibrational spectroscopy experiments. Quantum Monte Carlo techniques are correlated electronic structure methods promising for large molecules, which are intrinsically affected by stochastic errors on both energy and force calculations, making the mentioned calculations more challenging with respect to other more traditional quantum chemistry tools. To circumvent this drawback in the present work, we formulate the general problem of evaluating the molecular equilibrium structures, the harmonic frequencies, and the anharmonic coefficients of an error affected potential energy surface. The proposed approach, based on a multidimensional fitting procedure, is illustrated together with a critical evaluation of systematic and statistical errors. We observe that the use of forces instead of energies in the fitting procedure reduces the statistical uncertainty of the vibrational parameters by 1 order of magnitude. Preliminary results based on variational Monte Carlo calculations on the water molecule demonstrate the possibility to evaluate geometrical parameters and harmonic and anharmonic coefficients at this level of theory with an affordable computational cost and a small stochastic uncertainty (<0.07% for geometries and <0.7% for vibrational properties).

A second-order perturbation theory route to vibrational averages and transition properties of molecules: general formulation and application to infrared and vibrational circular dichroism spectroscopies

A general formulation to compute anharmonic vibrational averages and transition properties at the second-order of perturbation theory is derived from the Rayleigh-Schrödinger development. This approach is intended to be applicable to any property expanded as a Taylor series up to the third order with respect to normal coordinates or their associated momenta. The equations are straightforward to implement and can be easily adapted to various properties, as illustrated for the case of electric and magnetic dipole moments. From those, infrared and vibrational circular dichroism spectra can be readily obtained. This fully automatic procedure has been applied to several chiral molecules of small-to-medium sizes and compared to the standard double harmonic approximation and to experimental data.

Using neural networks, optimized coordinates, and high-dimensional model representations to obtain a vinyl bromide potential surface

We demonstrate that it is possible to obtain good potentials using high-dimensional model representations (HDMRs) fitted with neural networks (NNs) from data in 12 dimensions and 15 dimensions. The HDMR represents the potential as a sum of lower-dimensional functions and our NN-based approach makes it possible to obtain all of these functions from one set of fitting points. To reduce the number of terms in the HDMR, we use optimized redundant coordinates. By using exponential neurons, one obtains a potential in sum-of-products form, which greatly facilitates quantum dynamics calculations. A 12-dimensional (reference) potential surface for vinyl bromide is first refitted to show that it can be represented as a sum of two-dimensional functions. To fit 3d functions of the original coordinates, to improve the potential, a huge amount of data would be required. Redundant coordinates avoid this problem. They enable us to bypass the combinatorial explosion of the number of terms which plagues all HDMR and multimode-type methods. We also fit to a set of approximately 70,000 ab initio points for vinyl bromide in 15 dimensions [M. Malshe et al., J. Chem. Phys. 127, 134105 (2007)] and show that it is possible to obtain a surface in sum-of-products form of quality similar to the quality of the full-dimensional fit. Although we obtain a full-dimensional surface, we limit the cost of the fitting by building it from fits of six-dimensional functions, each of which requires only a small NN.

Vibrational spectroscopy of (SO42−)∙(H2O)n clusters, n=1–5: Harmonic and anharmonic calculations and experiment

The vibrational spectroscopy of (SO4(2-)).(H2O)n is studied by theoretical calculations for n=1-5, and the results are compared with experiments for n=3-5. The calculations use both ab initio MP2 and DFT/B3LYP potential energy surfaces. Both harmonic and anharmonic calculations are reported, the latter with the CC-VSCF method. The main findings are the following: (1) With one exception (H2O bending mode), the anharmonicity of the observed transitions, all in the experimental window of 540-1850 cm(-1), is negligible. The computed anharmonic coupling suggests that intramolecular vibrational redistribution does not play any role for the observed linewidths. (2) Comparison with experiment at the harmonic level of computed fundamental frequencies indicates that MP2 is significantly more accurate than DFT/B3LYP for these systems. (3) Strong anharmonic effects are, however, calculated for numerous transitions of these systems, which are outside the present observation window. These include fundamentals as well as combination modes. (4) Combination modes for the n=1 and n=2 clusters are computed. Several relatively strong combination transitions are predicted. These show strong anharmonic effects. (5) An interesting effect of the zero point energy (ZPE) on structure is found for (SO4(2-)).(H2O)(5): The global minimum of the potential energy corresponds to a C(s) structure, but with incorporation of ZPE the lowest energy structure is C2v, in accordance with experiment. (6) No stable structures were found for (OH-).(HSO4-).(H2O)n, for n<or=5.

The calculated infrared spectrum of Cl-H2O using a new full dimensional ab initio potential surface and dipole moment surface

We report a full dimensional, ab initio-based global potential energy surface (PES) and dipole moment surface for Cl-H2O. Both surfaces are symmetric with respect to interchange of the H atoms. The PES is a fit to thousands of electronic energies calculated using the coupled-cluster method [CCSD(T)] with a moderately large basis (aug-cc-pVTZ). Vibrational energies and wave functions are accurately obtained using MULTIMODE. The wave function and dipole moment surface are used to calculate and analyze the pure infrared spectrum at 0 K which is compared with experiment. Vibrational energies and the infrared spectra for DOD and HOD/DOH are also presented.

Using redundant coordinates to represent potential energy surfaces with lower-dimensional functions

We propose a method for fitting potential energy surfaces with a sum of component functions of lower dimensionality. This form facilitates quantum dynamics calculations. We show that it is possible to reduce the dimensionality of the component functions by introducing new and redundant coordinates obtained with linear transformations. The transformations are obtained from a neural network. Different coordinates are used for different component functions and the new coordinates are determined as the potential is fitted. The quality of the fits and the generality of the method are illustrated by fitting reference potential surfaces of hydrogen peroxide and of the reaction OH+H(2)-->H(2)O+H.

Anharmonic vibrational properties by a fully automated second-order perturbative approach

This paper describes the implementation of a fully automated code for the building of anharmonic force constants and their use in a second-order perturbative evaluation of vibrorotational parameters. Next, a number of test applications are discussed, which show the strengths and limits of various computational levels.

Efficient correlation-corrected vibrational self-consistent field computation of OH-stretch frequencies using a low-scaling algorithm

The authors present a new computational scheme to perform accurate and fast direct correlation-corrected vibrational self-consistent field (CC-VSCF) computations for a selected number of vibrational modes, which is aimed at predicting a few vibrations in large molecular systems. The method is based on a systematic selection of vibrational mode-mode coupling terms, leading to the direct ab initio construction of a sparse potential energy surface. The computational scaling of the CC-VSCF computation on the generated surface is then further reduced by using a screening procedure for the correlation-correction contributions. The proposed method is applied to the computation of the OH-stretch frequency of five aliphatic alcohols. The authors investigate the influence of different pseudopotential and all-electron basis sets on the quality of the correlated potential energy surfaces computed and on the OH-stretch frequencies calculated for each surface. With the help of these test systems, the authors show that their method offers a computational scaling that is two orders of magnitude lower than a standard CC-VSCF method and that it is of equal accuracy.

Ab Initio Study of the Isomeric Equilibrium of the HCN···H2O and H2O···HCN Hydrogen-Bonded Clusters

An ab initio study of the stability, spectroscopic properties, and isomeric equilibrium of the hydrogen-bonded HCN...H2O and H2O...HCN isomers is presented. Density functional theory and perturbative second-order MP2 and coupled-cluster CCSD(T) calculations were carried out and binding energies obtained with correlation-consistent basis sets including extrapolation to the infinity basis set level. At the best theoretical level, CCSD(T), the H2O...HCN complex is more stable than the HCN...H2O complex by ca. 6.3 kJ mol(-1). Rotational and vibrational spectra, including anharmonic corrections, are calculated. These calculated spectroscopic data are used to obtain thermochemical contributions to the thermodynamic functions and hence the Gibbs free energy. The relative free energies are used to estimate the equilibrium constant for isomerism. We find that under typical conditions of supersonic expansion experiments (T < 150 K) H2O...HCN is essentially the only isomer present. Furthermore, our calculations indicate that the hydrogen-bonded cluster becomes favorable over the separated moieties at temperatures below 200 K.

Converged vibrational energy levels and quantum mechanical vibrational partition function of ethane

The vibrational partition function of ethane is calculated in the temperature range of 200-600 K using well-converged energy levels that were calculated by vibrational configuration interaction, and the results are compared to the harmonic oscillator partition function. This provides the first test of the harmonic oscillator approximation for a molecule with more than five atoms. The absolute free energies computed by the harmonic oscillator approximation are in error by 0.59-0.62 kcal/mol over the 200-600 K temperature range.

Prediction of accurate anharmonic experimental vibrational frequencies for water clusters, (H2O)n, n=2-5

Accurate anharmonic experimental vibrational frequencies for water clusters consisting of 2-5 water molecules have been predicted on the basis of comparing different methods with MP2/aug-cc-pVTZ calculated and experimental anharmonic frequencies. The combination of using HF/6-31G* scaled frequencies for intramolecular modes and anharmonic frequencies for intermolecular modes gives excellent agreement with experiment for the water dimer and trimer and are as good as the expensive anharmonic MP2 calculations. The water trimer, the cyclic Ci and S4 tetramers, and the cyclic pentamer all have unique peaks in the infrared spectrum between 500 and 800 cm-1 and between 3400 and 3700 cm-1. Under the right experimental conditions these different clusters can be uniquely identified using high-resolution IR spectroscopy.

Experimental infrared spectra of Cl−(ROH) (R = H, CH3, CH3CH2) complexes in the gas-phase

Infrared multiple photon dissociation spectra for the chloride ion solvated by either water, methanol or ethanol have been recorded using an FTICR spectrometer coupled to a free-electron laser, and are presented here along with assignments to the observed bands. The assignments made to the Cl(-)/H(2)O, Cl(-)(CH(3)OH), and Cl(-)(CH(3)CH(2)OH) spectra are based on comparison with the neutral H(2)O, CH(3)OH, and CH(3)CH(2)OH spectra, respectively. This work confirms that a band observed around 1400 cm(-1) in the Cl(-)(H(2)O) spectrum is not due to the Ar tag in Ar predissociation spectra. The carrier of this band is, most likely, the first overtone of the OHCl bend. Based on the position of the overtone in the IRMPD spectrum, 1375 cm(-1), the fundamental must occur very close to 700 cm(-1) and observation of this band should aid theoretical treatments of the spectrum of this complex. B3LYP/6-311++G(2df,2pd) calculations are shown to reproduce the IRMPD spectra of all three solvated chloride species. They also predict that attaching one or two Ar atoms to the Cl(-)(H(2)O) complex results in a shift of no more than a few wavenumbers in the fundamental bands for the bare complex, in agreement with previous experiment. For both alcohol-Cl(-) complexes, the S(N)2 "backside attack" isomers are not observed and Cl(-) is predicted theoretically, and confirmed experimentally, to be bound to the hydroxyl hydrogen. For Cl(-)(CH(3)CH(2)OH), the trans and gauche conformers are similar in energy, with the gauche conformer predicted to be thermodynamically favoured. The experimental infrared spectrum agrees well with that predicted for the gauche conformer but a mixture of gauche and anti conformers cannot be ruled out based on the experimental spectra nor on the computed thermochemistry.

Global Search for Minimum Energy (H2O)n Clusters, n = 3−5

The Gaussian-3 (G3) model chemistry method has been used to calculate the relative deltaG(o) values for all possible conformers of neutral clusters of water, (H2O)n, where n = 3-5. A complete 12-fold conformational search around each hydrogen bond produced 144, 1728, and 20,736 initial starting structures of the water trimer, tetramer, and pentamer. These structures were optimized with PM3, followed by HF/6-31G* optimization, and then with the G3 model chemistry. Only two trimers are present on the G3 potential energy hypersurface. We identified 5 tetramers and 10 pentamers on the potential energy and free-energy hypersurfaces at 298 K. None of these 17 structures were linear; all linear starting models folded into cyclic or three-dimensional structures. The cyclic pentamer is the most stable isomer at 298 K. On the basis of this and previous studies, we expect the cyclic tetramers and pentamers to be the most significant cyclic water clusters in the atmosphere.

Ab Initio Vibrational Calculations for H2SO4 and H2SO4·H2O: Spectroscopy and the Nature of the Anharmonic Couplings

Vibrational frequencies for fundamental, overtone, and combination excitations of sulfuric acid (H2SO4) and of sulfuric acid monohydrate cluster (H2SO4 x H2O) are computed directly from ab initio MP2/TZP potential surface points using the correlation-corrected vibrational self-consistent field (CC-VSCF) method, which includes anharmonic effects. The results are compared with experiment. The computed transitions show in nearly all cases good agreement with experimental data and consistent improvement over the harmonic approximation. The CC-VSCF improvements over the harmonic approximation are largest for the overtone and combination excitations and for the OH stretching fundamental. The agreement between the calculations and experiment also supports the validity of the MP2/TZP potential surfaces. Anharmonic coupling between different vibrational modes is found to significantly affect the vibrational frequencies. Analysis of the mean magnitude of the anharmonic coupling interactions between different pairs of normal modes is carried out. The results suggest possible mechanisms for the internal flow of vibrational energy in H2SO4 and H2SO4 x H2O.

Calculations of Vibrational Energy Levels by Using a Hybrid ab Initio and DFT Quartic Force Field: Application to Acetonitrile

A hybrid quartic force field with quadratic force constants calculated at the CCSD(T)/cc-pVTZ level and cubic and quartic terms determined by a B3LYP/cc-pVTZ treatment is proposed to compute the vibrational energy levels of acetonitrile from a variational method. Fundamentals and overtones calculated in the range of 300-3200 cm(-1) are in fair agreement with the 31 observed data, with an absolute mean deviation of less than 0.8%. These results allow us to explain several Fermi resonances.

On the Contribution of Vibrational Anharmonicity to the Binding Energies of Water Clusters

The second-order vibrational perturbation theory method has been used together with the B3LYP and MP2 electronic structure methods to investigate the effects of anharmonicity on the vibrational zero-point energy (ZPE) contributions to the binding energies of (H2O)n, n = 2-6, clusters. For the low-lying isomers of (H2O)6, the anharmonicity correction to the binding energy is calculated to range from -248 to -355 cm(-1). It is also demonstrated that although high-order electron correlation effects are important for the individual vibrational frequencies, they are relatively unimportant for the net ZPE contributions to the binding energies of water clusters.

Influence of Substituents on the Anharmonicity of νs(OH) Vibration in Phenol Derivatives Explored by Experimental and Theoretical Approach

Very good reproducibility of the first five vibrational transitions of phenol in the gas phase by the MP2/ 6-31G potential for O-H bond stretching was found. The vibrational levels were calculated by a program for variational solving of the time-independent Schrödinger equation in one dimension. Relative intensities of particular transitions were determined on the basis of the function of the dipole moment. The substituent effects on the nu(s)(OH) transitions and on the intensity of these transitions, as well as on the structure of eleven phenols, was analyzed as a function of the pK(a) values.

Accurate vibrational-rotational partition functions and standard-state free energy values for H2O2 from Monte Carlo path-integral calculations

Accurate quantum mechanical partition functions and absolute free energies of H(2)O(2) are determined using a realistic potential energy surface [J. Koput, S. Carter, and N. C. Handy, J. Phys. Chem. A 102, 6325 (1998)] for temperatures ranging from 300 to 2,400 K by using Monte Carlo path integral calculations with new, efficient polyatomic importance sampling methods. The path centroids are sampled in Jacobi coordinates via a set of independent ziggurat schemes. The calculations employed enhanced-same-path extrapolation of trapezoidal Trotter Fourier path integrals, and the paths were constructed using fast Fourier sine transforms. Importance sampling was also used in Fourier coefficient space, and adaptively optimized stratified sampling was used in configuration space. The free energy values obtained from the path-integral calculations are compared to separable-mode approximations, to the Pitzer-Gwinn approximation, and to values in thermodynamic tables. Our calculations support the recently proposed revisions to the JANAF tables.

Vibrational zero-point energies and thermodynamic functions beyond the harmonic approximation

This paper compares harmonic and anharmonic zero-point energies and thermodynamic functions for a number of molecules of small and medium size. Anharmonic corrections cannot be neglected for quantitative studies, but can be obtained quite effectively by a perturbative treatment including cubic force constants to the second order and semidiagonal quartic constants to the first order. Simple finite difference equations provide all the necessary terms by at most 6N-11 Hessian evaluations, where N is the number of atoms in the system. Accurate values are obtained by this method using the Becke three parameter Lee-Yang-Parr functional, medium size basis sets, and, when needed, proper treatment of internal rotations. The whole model has been completely automated in the Gaussian package.

Fast vibrational self-consistent field calculations through a reduced mode–mode coupling scheme

We present a new methodology to perform fast correlation-corrected vibrational self-consistent field (CC-VSCF) calculations using ab initio potential energy points calculated on the fly. Our method is based on the replacement of all-electron basis sets with a pseudo-potential basis for heavy atoms, and on an efficient reduction of the number of pair-coupling elements used in the CC-VSCF procedure. The method is applied to several test systems: H2O, NH3, and CH4, where it proves to be efficient, providing a speedup factor of 2 compared to a standard CC-VSCF calculation. We also apply our technique to the simulation of the vibrational spectrum of ethane and show that very accurate results can be obtained with a substantial speedup for this system.