Vibrational Self-consistent Field Method Research Articles

Vibrational Self-Consistent Field (VSCF) and Post-VSCF Method Calculations Combined with the Reference Interaction Site Model Self-Consistent Field Method Coupled with the Constrained Spatial Electron Density Distribution: Applications to NaHCOO in Aqueous Phase.

Investigating vibrational behavior in solution is crucial for understanding molecular dynamics within a solvent environment. Notably, the analysis of Raman spectra for molecules in solution is important owing to its ability to unveil intricate solute-solvent interactions. Previous studies have effectively employed frequency calculations utilizing the reference interaction site model self-consistent field method in conjunction with constrained spatial electron density distribution (RISM-SCF-cSED) to understand molecular vibrations in solution, primarily focusing on fundamental vibrational modes. However, the oversight of overtones and combination tones in these studies prompted us to combine the vibrational self-consistent field (VSCF) and vibrational second-order Mo̷ller-Plesset perturbation (VMP2) methods with RISM-SCF-cSED to address these aspects theoretically. Illustrating the efficacy of this integrated approach, we computed the Raman spectra of sodium formate (NaHCOO) in water, revealing the necessity of accounting for molecular anharmonicity in solution vibrational analysis. Our findings underscore the potency of VSCF and VMP2 in conjunction with RISM-SCF-cSED as a robust theoretical framework for such calculations.

First-Principles Anharmonic Infrared and Raman Vibrational Spectra of Materials: Fermi Resonance in Dry Ice.

We introduce a computational tool for the quantum-mechanical simulation of anharmonic infrared and Raman vibrational spectra of materials. The approach, implemented in the CRYSTAL software, stems from Taylor's expansion of the potential energy surface (PES) on the basis of normal modes up to cubic and quartic terms. The PES can be sampled with four different numerical schemes at the level of density functional theory (DFT), with local, generalized-gradient, and hybrid density functional approximations. Anharmonic states are obtained by solving Shrödinger's nuclear equation with either the vibrational self-consistent field (VSCF) or vibrational configuration interaction (VCI) methods. Nuclear quantum effects (NQEs) are thus fully accounted for. Infrared intensities are computed numerically through a Berry phase approach or analytically through a coupled-perturbed (CP) approach. Raman intensities are computed analytically via the CP approach. A variety of anharmonic features of vibrational spectra of materials can be simulated, including band shifts, combination bands, overtones, resonances (first-order Fermi, second-order Darling-Dennison), and hot bands. We showcase the effectiveness of the approach on the description of a first-order Fermi resonance (FR) in CO2 dry ice: a challenging test-case given that the FR occurs in the Raman spectrum, requires NQEs, and involves two- and three-mode couplings. Fundamental mechanistic differences with respect to the well-known FR in molecular CO2 are addressed. This application represents the first quantum-mechanical, periodic description of FR in dry ice.

Accelerating and stabilizing the convergence of vibrational self-consistent field calculations via the direct inversion of the iterative subspace (vDIIS) algorithm.

The vibrational self-consistent field (VSCF) method yields anharmonic states and spectra for molecular vibrations, and it serves as the starting point for more sophisticated correlated-vibration methods. Convergence of the iterative, non-linear optimization in VSCF calculations can be erratic or altogether unsuccessful, particularly for chemical systems involving low-frequency motions. In this work, a vibrational formulation of the Direct Inversion of the Iterative Subspace method of Pulay is presented and investigated. This formulation accounts for distinct attributes of the vibrational and electronic cases, including the expansion of each single-mode vibrational wavefunction in its own basis set. The resulting Direct Inversion of the Iterative Subspace method is shown to substantially accelerate VSCF convergence in all convergent cases as well as rectify many cases where Roothaan-based methods fail. Performance across systems ranging from small, rigid molecules to weakly bound molecular clusters is investigated in this analysis.

Anharmonic Coupling of Stretching Vibrations in Ice: A Periodic VSCF and VCI Description.

The anharmonicity of O-H stretching vibrations of water ice is characterized by use of a periodic implementation of the vibrational self-consistent field (VSCF) and vibrational configuration interaction (VCI) methods, which take phonon-phonon couplings explicitly into account through numerical evaluation of high-order terms of the nuclear potential. The low-temperature, proton-ordered phase of water ice (namely, ice XI) is investigated. The net effect of a coupled anharmonic treatment of stretching modes is not just a rigid blue-shift of the respective harmonic spectral frequencies but rather a complex change of their relative spectral positions, which cannot be captured by simple scaling strategies based on harmonic calculations. The adopted techniques allow for a hierarchical treatment of anharmonic terms of the nuclear potential, which is key to an effective identification of leading factors. We show that the anharmonic independent-mode approximation─only describing the "intrinsic anharmonicity" of the O-H stretches─is unable to capture the correct physics, and that couplings among O-H stretches must be described. Inspection of harmonic normal coordinates allows identification of specific features of the O-H stretching motions which most likely enable strong mode-mode couplings. Finally, by coupling O-H stretches to all other possible modes of ice XI (THz collective vibrations, molecular librations, bendings), we identify specific types of motion which significantly affect O-H stretching states: in particular, molecular librations are found to affect the stretching states more than molecular bendings.

Calculation of absolute Raman scattering cross-sections using vibrational self-consistent field/vibrational configuration interaction wave functions.

In the present study, the differential scattering cross-sections, depolarization ratios and Raman shifts of small molecular systems are obtained from configuration iteration wave functions of vibrational self-consistent field (VSCF) states. The transition polarizabilities were modeled using the Placzek approximation, neglecting those contributions not arising from the electric dipole mechanism. This theoretical approach is considered a good approximation for samples that absorb in the UV range if the excitation radiation falls in the visible region, as is the case of the molecules selected for the present study, namely: water, methane, and acetylene. Potential energy and electronic polarizability surfaces are calculated by the CCSD(T) and CC3 methods with aug-cc-p(C)V(T,Q,5)Z basis sets. The vibrational Hamiltonian includes the vibrational angular momentum contribution of the Watson kinetic energy operator. As expected, due to the variational nature of the VSCF and vibrational configuration interaction (VCI) methods, the Raman transition wavenumbers are substantially improved over the harmonic predictions. Surprisingly, the scattering cross-sections obtained using the harmonic approximation or the VSCF method better agrees with the experimental values than those cross-sections predicted using VCI wave functions. The more significant deviations of the VCI results from the experimental reference may be related to the significant uncertainties of the measured cross-sections. Still, it may also indicate that the VCI Raman transition moments may require a more accurate description of the electronic polarizability surface. Finally, the depolarization ratios calculated for H2 O and C2 D2 using harmonic and VCI wave functions have similar accuracy, whereas, for C2 H2 and C2 HD, the VCI results are more accurate.

Performance of Vibrational Self-Consistent Field Theory for Accurate Potential Energy Surfaces: Fundamentals, Excited States, and Intensities.

The performance of vibrational structure calculations beyond harmonic approximation in the framework of the vibrational self-consistent field method with second-order perturbation corrections (VSCF-PT2) is investigated in conjunction with very accurate potential energy surfaces (PESs) given by various coupled-cluster electronic structure theories. The quality of anharmonic calculations depends on the accuracy of the underlying multidimensional PES obtained from its functional form, which is given by the level of electronic structure theory. Two such highest levels of typical coupled-cluster electronic structure methods, CCSD and the ″gold standard″ CCSD(T), along with their variants such as CCD, CR-CCL (completely renormalized CR-CC(2,3) approach), and CCSD(TQ) are tested for the construction of accurate anharmonic potentials without any fitting or ad hoc scaling and using cc-pVTZ basis sets. The accuracy of VSCF-PT2 theory in comparison to experimental values is tested for a series of 16 molecules with 135 fundamental bands, 64 overtones, and combination bands and also for 39 intensities. It is found that CCD and CCSD bind the potential tighter than CCSD(T) and the computed VSCF-PT2 transitions are more blue-shifted showing higher deviation from the experiment. In general, VSCF-PT2 results computed at the CCSD(T) potential offer a good cost/accuracy ratio, with the mean absolute deviation and the mean absolute percentage error with the experiment being ∼16 cm-1 and 1.38, respectively, for fundamentals. Additionally, while the CR-CCL and CCSD(TQ) methods offer similar levels of accuracies as compared to CCSD(T), the former offers a better accuracy/cost ratio than the latter and is a suitable alternative to CCSD(T).

Anharmonic vibrational computations with a quartic force field for curvilinear coordinates

The direct vibrational self-consistent field (VSCF) method, which combines anharmonic vibrational theory with electronic structure calculations, is a sophisticated theoretical approach to calculate the vibrational spectra of molecules from first principles. Combining the VSCF approach with the quartic force field (QFF) is a good alternative to direct VSCF, with a lower computational cost. QFF is a 4th-order Taylor expansion of the potential energy surface near an equilibrium geometry. In this study, a new strategy is proposed to derive the QFF in terms of normal coordinates; the QFF coefficients are determined through numerical differentiations of the energy by representing the normal coordinates in internal rather than Cartesian coordinates. The VSCF/QFF-internal method was implemented in the General Atomic and Molecular Electronic Structure System electronic structure program and applied to the evaluations of the fundamental vibrational frequencies of HNO2, HNO3, H2O dimer, and H2O trimer, using Møller-Plesset second order perturbation theory and the aug-cc-pVDZ and aug-cc-pVTZ basis sets. The results are much improved, especially for the intermolecular vibrational modes, compared with the Cartesian coordinate representation of the normal coordinates in the VSCF/QFF approach.

Anharmonic Vibrational States of Solids from DFT Calculations. Part II: Implementation of the VSCF and VCI Methods.

Two methods are implemented in the Crystal program for the calculation of anharmonic vibrational states of solids: the vibrational self-consistent field (VSCF) and the vibrational configuration-interaction (VCI). While the former is a mean-field approach, where each vibrational mode interacts with the average potential of the others, the latter allows for an explicit and complete account of mode-mode correlation. Both schemes are based on the representation of the adiabatic potential energy surface (PES) discussed in Part I, where the PES is expanded in a Taylor's series so as to include up to cubic and quartic terms. The VSCF and VCI methods are formally presented and their numerical parameters discussed. In particular, the convergence of computed anharmonic vibrational states, within the VCI method, is investigated as a function of the truncation of the expansion of the nuclear wave function. The correctness and effectiveness of the implementation is discussed by comparing with available theoretical and experimental data on both molecular and periodic systems. The effect of the adopted basis set and exchange-correlation functional in the description of the PES on computed anharmonic vibrational states is also addressed.

First-principles anharmonic vibrational study of the structure of calcium silicate perovskite under lower mantle conditions

Calcium silicate perovskite (CaSiO$_3$) is one of the major mineral components of the lower mantle, but has been the subject of relatively little work compared to the more abundant Mg-based materials. One of the major problems related to CaSiO$_3$ that is still the subject of research is its crystal structure under lower mantle conditions - a cubic Pm$\bar{3}$m structure is accepted in general, but some have suggested that lower-symmetry structures may be relevant. In this paper, we use a fully first-principles vibrational self-consistent field method to perform high accuracy anharmonic vibrational calculations on several candidate structures at a variety of points along the geotherm near the base of the lower mantle to investigate the stability of the cubic structure and related distorted structures. Our results show that the cubic structure is the most stable throughout the lower mantle, and that this result is robust against the effects of thermal expansion.

High-dimensional fitting of sparse datasets of CCSD(T) electronic energies and MP2 dipole moments, illustrated for the formic acid dimer and its complex IR spectrum.

We present high-level, coupled-mode calculations of the infrared spectrum of the cyclic formic acid dimer. The calculations make use of full-dimensional, ab initio potential energy and dipole moment surfaces. The potential is a linear least-squares fit to 13 475 CCSD(T)-F12a/haTZ (haTZ means aug-cc-pVTZ basis set for O and C, and cc-pVTZ for H) energies, and the dipole moment surface is a fit to the dipole components, calculated at the MP2/haTZ level of theory. The variables of both fits are all (45) internuclear distances (actually Morse variables). The potential, which is fully permutationally invariant, is the one published recently and the dipole moment surface is newly reported here. Details of the fits, especially the dipole moment, and the database of configurations are given. The infrared spectrum of the dimer is calculated by solving the nuclear Schrödinger equation using a vibrational self-consistent field and virtual-state configuration interaction method, with subsets of the 24 normal modes, up to 15 modes. The calculations indicate strong mode-coupling in the C-H and O-H stretching region of the spectrum. Comparisons are made with experiments and the complexity of the experimental spectrum in the C-H and O-H stretching region is successfully reproduced.

Synthesis, characterization and vibrational studies of p-chlorosulfinylaniline

p-Cholorosulfinylaniline was prepared by the reaction of p-chloroaniline and SOCl2. The structural, conformational and configurational properties of the obtained liquid compound were studied by Raman and infrared spectroscopy in the liquid state. The assignment of the vibrational spectra was carried out with the help of data obtained by quantum chemical calculations at the harmonic oscillator approximation and using anharmonic vibrational self-consistent field (VSCF) method as well. The 1H and 13C NMR chemical shifts of the molecule were calculated by Gauge including orbital (GIAO) method (DFT/B3LYP approximation using 6-311 + G (df), 6-311++G (df,pd) and cc-pVTZ basis sets) and compared to the experimental values. Natural Bond Orbital analysis provides an explanation of the stability of the molecule and its electronic properties upon charge delocalization.

Vibrational and electronic spectral analysis of 2,3-pyrazinedicarboxylic acid: A combined experimental and theoretical study

ABSTRACTFourier transform infrared and Raman spectra of 2,3-pyrazinedicarboxylic acid were recorded and analyzed using density functional theory. The complete assignments of the anharmonic vibrational modes have been performed based on potential energy distribution. The anharmonic frequencies were computed using vibrational second-order perturbation theory as well as vibrational self-consistent field and correlation corrected vibrational self-consistent field methods. Mode–mode coupling strength is also estimated using two-mode representation of quartic force field approximation. The intra- and intermolecular interactions were also studied in the dimer and trimer forms of the title molecule. The ultraviolet–visible absorption spectra in ethanol, methanol, and acetonitrile solvents were recorded and analyzed using time-dependent density functional theory involving the polarization continuum model. The observed and calculated results are well comparable. Molecular electrostatic potential and the highest occupied and the lowest unoccupied molecular orbital analyses are also reported.

Infrared Spectrum of Toluene: Comparison of Anharmonic Isolated-Molecule Calculations and Experiments in Liquid Phase and in a Ne Matrix

First-principles anharmonic calculations are carried out for the CH stretching vibrations of isolated toluene and compared with the experimental infrared spectra of isotopologues of toluene in a Ne matrix at 3 K and of liquid toluene at room temperature. The calculations use the vibrational self-consistent field method and the B3LYP potential surface. In general, good agreement is found between the calculations and experiments. However, the spectrum of toluene in a Ne matrix is more complicated than that predicted theoretically. This distinction is discussed in terms of matrix-site and resonance effects. Interestingly, the strongest peak in the CH stretching spectrum has similar widths in the liquid phase and in a Ne matrix, despite the very different temperatures. Implications of this observation to the broadening mechanism are discussed. Finally, our results show that the B3LYP potential offers a good description of the anharmonic CH stretching band in toluene, but a proper description of matrix-site and resonance effects remains a challenge.

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. We report harmonic, VSCF, and correlation-corrected VSCF calculations obtained using a vibrational configuration interaction approach (VSCF/VCI). Our implementation of the method is based on an efficient calculation of the potential energy surfaces (PES), using periodic density functional theory (DFT) with a plane-wave pseudopotential basis. In some cases, we use an efficient technique (fast-VSCF) assisted by the Voter-Chen potential in order to get an efficient reduction of the number of pair-couplings between modes. This allows us to efficiently reduce the computing time of 2D-PES without degrading the accuracy. We found that anharmonicity of the gold clusters is very small with maximum rms deviations of about 1 cm(-1), although for some particular modes anharmonicity reaches values slightly larger than 2 cm(-1). Silver clusters show slightly larger anharmonicity. In both cases, large differences between calculated and experimental vibrational frequencies (when available) stem more likely from the quality of the electronic structure method used than from vibrational anharmonicity. We show that noble gas embedding often affects the vibrational properties of these clusters more than anharmonicity, and discuss our results in the context of experimental studies.

An ab initio potential energy surface for the formic acid dimer: zero-point energy, selected anharmonic fundamental energies, and ground-state tunneling splitting calculated in relaxed 1-4-mode subspaces.

We report a full-dimensional, permutationally invariant potential energy surface (PES) for the cyclic formic acid dimer. This PES is a least-squares fit to 13475 CCSD(T)-F12a/haTZ (VTZ for H and aVTZ for C and O) energies. The energy-weighted, root-mean-square fitting error is 11 cm-1 and the barrier for the double-proton transfer on the PES is 2848 cm-1, in good agreement with the directly-calculated ab initio value of 2853 cm-1. The zero-point vibrational energy of 15 337 ± 7 cm-1 is obtained from diffusion Monte Carlo calculations. Energies of fundamentals of fifteen modes are calculated using the vibrational self-consistent field and virtual-state configuration interaction method. The ground-state tunneling splitting is computed using a reduced-dimensional Hamiltonian with relaxed potentials. The highest-level, four-mode coupled calculation gives a tunneling splitting of 0.037 cm-1, which is roughly twice the experimental value. The tunneling splittings of (DCOOH)2 and (DCOOD)2 from one to three mode calculations are, as expected, smaller than that for (HCOOH)2 and consistent with experiment.

Towards an automated and efficient calculation of resonating vibrational states based on state-averaged multiconfigurational approaches.

Resonating vibrational states cannot consistently be described by single-reference vibrational self-consistent field methods but request the use of multiconfigurational approaches. Strategies are presented to accelerate vibrational multiconfiguration self-consistent field theory and subsequent multireference configuration interaction calculations in order to allow for routine calculations at this enhanced level of theory. State-averaged vibrational complete active space self-consistent field calculations using mode-specific and state-tailored active spaces were found to be very fast and superior to state-specific calculations or calculations with a uniform active space. Benchmark calculations are presented for trans-diazene and bromoform, which show strong resonances in their vibrational spectra.

Stochastic algorithm for size-extensive vibrational self-consistent field methods on fully anharmonic potential energy surfaces.

A stochastic algorithm based on Metropolis Monte Carlo (MC) is presented for the size-extensive vibrational self-consistent field methods (XVSCF(n) and XVSCF[n]) for anharmonic molecular vibrations. The new MC-XVSCF methods substitute stochastic evaluations of a small number of high-dimensional integrals of functions of the potential energy surface (PES), which is sampled on demand, for diagrammatic equations involving high-order anharmonic force constants. This algorithm obviates the need to evaluate and store any high-dimensional partial derivatives of the potential and can be applied to the fully anharmonic PES without any Taylor-series approximation in an intrinsically parallelizable algorithm. The MC-XVSCF methods reproduce deterministic XVSCF calculations on the same Taylor-series PES in all energies, frequencies, and geometries. Calculations using the fully anharmonic PES evaluated on the fly with electronic structure methods report anharmonic effects on frequencies and geometries of much greater magnitude than deterministic XVSCF calculations, reflecting an underestimation of anharmonic effects in a Taylor-series approximation to the PES.

Normal-ordered second-quantized Hamiltonian for molecular vibrations.

A normal-ordered second-quantized form of the Hamiltonian is derived for quantum dynamics in a bound potential energy surface expressed as a Taylor series in an arbitrary set of orthogonal, delocalized coordinates centered at an arbitrary geometry. The constant, first-, and second-order excitation amplitudes of this Hamiltonian are identified as the ground-state energy, gradients, and frequencies, respectively, of the size-extensive vibrational self-consistent field (XVSCF) method or the self-consistent phonon method. They display the well-defined size dependence of V(1-n/2), where V is the volume and n is the number of coordinates associated with the amplitudes. It is used to rapidly derive the equations of XVSCF and vibrational many-body perturbation methods with the Møller-Plesset partitioning of the Hamiltonian.

Efficient anharmonic vibrational spectroscopy for large molecules using local-mode coordinates.

This article presents a general computational approach for efficient simulations of anharmonic vibrational spectra in chemical systems. An automated local-mode vibrational approach is presented, which borrows techniques from localized molecular orbitals in electronic structure theory. This approach generates spatially localized vibrational modes, in contrast to the delocalization exhibited by canonical normal modes. The method is rigorously tested across a series of chemical systems, ranging from small molecules to large water clusters and a protonated dipeptide. It is interfaced with exact, grid-based approaches, as well as vibrational self-consistent field methods. Most significantly, this new set of reference coordinates exhibits a well-behaved spatial decay of mode couplings, which allows for a systematic, a priori truncation of mode couplings and increased computational efficiency. Convergence can typically be reached by including modes within only about 4 Å. The local nature of this truncation suggests particular promise for the ab initio simulation of anharmonic vibrational motion in large systems, where connection to experimental spectra is currently most challenging.

Dissociation of high-pressure solid molecular hydrogen: a quantum Monte Carlo and anharmonic vibrational study.

A theoretical study is reported of the molecular-to-atomic transition in solid hydrogen at high pressure. We use the diffusion quantum MonteCarlo method to calculate the static lattice energies of the competing phases and a density-functional-theory-based vibrational self-consistent field method to calculate anharmonic vibrational properties. We find a small but significant contribution to the vibrational energy from anharmonicity. A transition from the molecular Cmca-12 direct to the atomic I41/amd phase is found at 374GPa. The vibrational contribution lowers the transition pressure by 91GPa. The dissociation pressure is not very sensitive to the isotopic composition. Our results suggest that quantum melting occurs at finite temperature.