Zero-point Vibrational Corrections Research Articles

Heats of Formation of Alkali Metal and Alkaline Earth Metal Oxides and Hydroxides: Surprisingly Demanding Targets for High-Level ab Initio Procedures

High-level ab initio calculations, including variants of the Wn and G3 procedures, have been used to determine the structures and heats of formation of the alkali metal and alkaline earth metal oxides and hydroxides (M2O, MOH: M = Li, Na, and K; MO, M(OH)2: M = Be, Mg, and Ca). Our best structures were obtained at the CCSD(T)(riv,rv)/aug‘-cc-pWCVQZ level and are in uniformly close agreement with available experimental data, with a mean absolute deviation from experimental metal−oxygen bond lengths of just 0.007 A.Structures obtained with CCSD(T)/cc-pWCVQZ, B3-LYP/cc-pVTZ, B3-LYP/6-31G(2df,p), and MP2(full)/6-311+G(3df,2p) are also in good agreement with experiment. Zero-point vibrational energies and enthalpy temperature corrections are found to be relatively insensitive to the various procedures employed. However, the heats of formation for these molecules are challenging targets for high-level ab initio procedures. In the Wn-type procedures, it is found that expanding the correlation space on the meta...

Vibrational Effects on Molecular Properties in Large Molecules

We discuss an approach for calculating vibrational wave functions, eigenenergies and zero-point vibrational corrections to molecular properties. For simplicity the detailed theory is presented for a diatomic molecule, but the formulas for a general polyatomic molecule are also given. The approach has been implemented so as to allow vibrational corrections to be calculated for large molecules. We discuss recent applications of the model for calculating zero-point vibrational corrections to properties such as nuclear shielding constants, magnetizabilities, polarizabilities and optical rotation, and the use of this approach for obtaining accurate vibrational frequencies for bimolecular complexes.

Melting Temperature, Brillouin Shift, and Density of States of Nanocrystals

The amplitude of displacement of atoms due to scattering at the surface is calculated. It is found that the zero-point vibrational correction to the mean-square displacement varies as the inverse square of the nanocrystal size and the temperature-dependent part also varies with the nanocrystal size. The melting temperature is found to depend on the nanocrystal size so that smaller melting temperature is obtained for smaller nanocrystals. The phonon−phonon scattering is found to shift the Brillouin line such that as the crystal size reduces, it moves toward the Rayleigh line. The phonon density of states is found to be larger for nanocrystals than for the bulk crystals as it depends on the inverse square of the nanocrystal size.

Hybrid exchange-correlation functional determined from thermochemical data and ab initio potentials

Multiplicative potentials, appropriate for adding to the non-multiplicative fractional orbital exchange term in the Kohn–Sham equations, are determined from correlated ab initio electron densities. The potentials are examined graphically and are used in conjunction with conventional thermochemical data to determine a new hybrid exchange-correlation functional, denoted B97-2. Calculations using B97-2 are compared with those from (a) the B97-1 functional [J. Chem. Phys. 109, 6264 (1998)], which has the same functional form and fraction of orbital exchange, but was fitted to just thermochemical data; and (b) the widely used B3LYP functional [J. Chem. Phys. 98, 5648 (1993)]. B97-2 atomization energies are close to those from B97-1; total electronic energies and ionization potentials are less accurate, but remain an improvement over B3LYP. Molecular structures from all three functionals are comparable. Static isotropic polarizabilities improve from B3LYP to B97-1 to B97-2; the B97-2 functional underestimates experimental values, which is consistent with the neglect of zero-point vibrational corrections. NMR shielding constants—determined as the conventional second derivative of the electronic energy—improve from B3LYP to B97-1 to B97-2. Shieldings determined directly from these DFT electron densities using the recently proposed MKS approach [Chem. Phys. Lett. 337, 341 (2001)] are two to three times more accurate than the conventional shieldings, and exhibit an analogous improvement across the three functionals. Classical reaction barriers for sixteen chemical reactions improve significantly from B3LYP to B97-1 to B97-2. The introduction of multiplicative potentials into semi-empirical hybrid functional development therefore appears beneficial.

Molecular Magnetizabilities: Zero-Point Vibrational Effects and the Breakdown of Pascal's Rule

We demonstrate, by ab initio calculations on more than 60 molecules, that zero-point vibrational corrections to isotropic magnetizabilities in general are negligible, being less than 0.5% for almost all molecules studied. The exceptions to this rule are aromatic and anti-aromatic ring systems where the effect may be as large as 1% due to a sizable vibrational contribution to the component of the magnetizability perpendicular to the molecular plane. For the magnetizability anisotropy, zero-point vibrational corrections are much more important, often contributing 5−10% of the total vibrationally averaged magnetizability anisotropy. We also demonstrate that the additivity of magnetizabilities (known as Pascal's rule) breaks down in the case of fluorine-containing molecules.

Vibrational corrections to electric properties of weakly bound systems

The pure electronic and vibrational contributions to electric dipole moments, dipole polarizabilities, and first hyperpolarizabilities have been evaluated for the HF and H2O dimers. The zero-point vibrational average corrections to dipole moments and dipole polarizabilities turn out to be relatively small. However, the corresponding contributions to the first hyperpolarizability are found to be of the same magnitude as the pure electronic values. The so-called pure vibrational corrections to the dipole polarizability and first hyperpolarizability of hydrogen bonded dimers are exceptionally large and indicate that the perturbation theory method used for their evaluation fails to account properly for the high mechanical and electric anharmonicities present in these systems. The analysis of different harmonic and anharmonic contributions to the pure vibrational correction to the first hyperpolarizability shows explicitly the importance of the low frequency intermolecular modes.

Zero-point vibrational effects on optical rotation

We investigate the effects of molecular vibrations on the optical rotation in two chiral molecules, methyloxirane and trans-2,3-dimethylthiirane. It is shown that the magnitude of zero-point vibrational corrections increases as the electronic contribution to the optical rotation increases. Vibrational effects thus appear to be important for an overall estimate of the molecular optical rotation, amounting to about 20–30% of the electronic counterpart. We also investigate the special case of chirality introduced in a molecule through isotopic substitution. In this case, the zero-point vibrational effects are the only source of optical activity, and we investigate the magnitude of these effects through calculations on mono-deuterated oxirane.

Tautomeric equilibria of 2- and 4-thiouracil in gas phase and in solvent: A density functional study

The relative stabilities of the five favored tautomers of 2- and 4-thiouracil in gas phase and in water solution were determined by density functional theory employing the Becke, Lee, Yang, and Parr (B3LYP) exchange–correlation potential and the three 6-31G(d,p), 6-311++G(d,p), and triple-zeta valence (TZVP) basis sets. Zero-point vibrational corrections were also computed. Bulk solvent effects were studied in the framework of the self-consistent reaction field approach by the polarizable continuum model. All calculations indicate that the most stable tautomer for both species, in the gas phase as well as in solution, has the oxo-thione form, in full agreement with the previous ab initio and experimental studies. The tautomeric stability orders obtained in the aqueous solution are sensibly different from that in the gas phase. At B3LYP/6-311++G(d,p) level in the gas phase, the following orders of stability for 2- and 4-thiouracil tautomers were observed, respectively: S2U1>S2U2>S2U4>S2U5>S2U3 and S4U1>S4U2>S4U3>S4U4>S4U5. The corresponding trends in the aqueous phase are S2U1>S2U3>S2U2>S2U5>S2U4 and S4U1>S4U2>S4U3>S4U5>S4U4. On the basis of the computed energy differences we can hypothesize that only the oxo-thione forms of 2- and 4-thiouracil should exist in the gas phase and in water solution. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 44–52, 2001

Determination of vibrational polarizabilities and hyperpolarizabilities using field-induced coordinates

An analytical set of field-induced coordinates (FICs) is defined. It is shown that, instead of 3N−6 normal coordinates, a relatively small number of FICs is sufficient to describe the vibrational polarizability and hyperpolarizabilities due to nuclear relaxation. The fact that the number of FICs does not depend upon the size of the molecule leads to computational advantages. A method is provided for separating anharmonic contributions from harmonic contributions as well as effective mechanical from electrical anharmonicity. Hartree–Fock calculations on a dozen representative conjugated molecules illustrate the procedures and indicate that anharmonicity can be very important. Other potential applications including the determination of zero-point vibrational averaging corrections are noted.

Accurate ab Initio Calculations on the Rate Constants of the Direct Hydrogen Abstraction Reaction C2H + H2 → C2H2 + H

The direct hydrogen abstraction process for the reaction C2H + H2 → C2H2 + H is theoretically investigated at the UQCISD/6-311+G(d,p) and G2//UQCISD levels. The reaction rate constants over a wide range of temperatures from 20−5000 K are calculated over a wide range of temperatures from 20−5000 K using the canonical variational transition state theory along with the small curvature tunneling correction. At the G2//UQCISD level without and with zero-point vibrational correction, the obtained forward reaction barrier are 2.22 and 2.52 kcal/mol, respectively, which, for the first time, well match the value 2.0−2.5 kcal/mol expected by many groups. The obtained forward rate constants are also in good agreement with most of the available experimental results covering a wide temperature range. Furthermore, the calculated reverse rate constants agree well with two recent experimental estimates yet differ from the other experimental estimates and one ab initio calculation at lower level. Finally, our calculated r...

An ab initio study of vibrational corrections to the electrical properties of ethylene

Accurate SCF and MP2 quartic property hypersurfaces have been computed for the energy, quadrupole moment and polarizability tensor of ethylene to obtain zero-point vibrational corrections to the properties. Coupled with accurate electrical properties computed at a high level correlated r e geometry, using a range of correlated methods, especially BD and BD(T), along with a number of purpose-built polarized basis sets, definitive estimates have been made of these properties that incorporate the effects of vibrational averaging. The effect of deuterium substitution on the properties was investigated, and the frequency dependence of the polarizability tensor was studied also. Careful attention has been paid to a critical comparison between these theoretical estimates and experimental measurements, and agreement between the two is shown to be exceptionally good. In particular, it is possible to resolve the disagreement between recent theoretical calculations and experimental measurements of the Cotton–Mouton constant. The results focus attention on both the general utility of the present method, and the necessity to allow for the effects of zero-point vibrational averaging when comparing theory with experiment, or even when comparing different theoretical results with one another using experiment as a benchmark.

An ab initio study of vibrational corrections to the electrical properties of ethane

SCF quartic property hypersurfaces have been computed for the energy, quadrupole moment and polarizability tensor of ethane via finite differences of analytical derivatives to obtain zeropoint vibrational corrections to the properties. Coupled with accurate electrical properties computed at a correlated r e geometry, using a range of correlated methods, especially BD and BD(T), excellent estimates of these properties have been obtained that incorporate the effects of vibrational averaging. The effect of deuterium substitution on the properties has been investigated, including the first known theoretical prediction of the dipole moment of CH3CD3, and the frequency dependence of the polarizability tensor and Rayleigh depolarization ratio have been examined. Careful attention has been paid to a critical comparison between these theoretical estimates and experimental measurements. Although the vibrational corrections are limited to SCF theory, and basis sets not as extensive as used in some studies of this kind, nevertheless the results can be compared quantitatively with experiment, and the agreement is found to be excellent. The results underline the necessity to allow for the effects of zero-point vibrational averaging when comparing theory with experiment, and suggest a viable approach for studies on larger molecules.

An ab initio study of vibrational corrections to the electrical properties of the fluoromethanes: CH3F, CH2F2, CHF3 and CF4

Accurate SCF and MP2 quartic property hypersurfaces have been computed for the energy, dipole moments, quadrupole moments and polarizability tensors of the fluorinated methanes CF4, CHF3, CH2F2 and CH3F, to establish accurate values of zero-point vibrational corrections to the properties. Using a consistent set of re geometries from density functional theory, these ZPVCs are coupled with accurate electrical properties computed using a range of correlated methods, especially BD and BD(T), and a number of purpose-built polarized basis sets, to obtain near definitive estimates of these properties that incorporate the effects of vibrational averaging. Careful attention has been paid to a critical comparison between these theoretical estimates and experimental measurements, and agreement between the two is shown to be exceptionally good. In particular, it is clear that in many instances more precise experimental results would be required in order to discriminate between different correlated results, or between the present results and those which may be obtained with larger basis sets. The work highlights the necessity to allow for the effects of zero-point vibrational averaging when comparing theory with experiment, or even when comparing different theoretical results with one another using experiment as a benchmark. It also points to the need for further precise experimental measurements of some of these properties.

An efficient approach for calculating vibrational wave functions and zero-point vibrational corrections to molecular properties of polyatomic molecules

We have recently presented a formalism for calculating zero-point vibrational corrections to molecular properties of polyatomic molecules in which the contribution to the zero-point vibrational correction from the anharmonicity of the potential is included in the calculations by performing a perturbation expansion of the vibrational wave function around an effective geometry. In this paper we describe an implementation of this approach, focusing on computational aspects such as the definition of normal coordinates at a nonequilibrium geometry and the use of the Eckart frame in order to obtain accurate nonisotropic molecular properties. The formalism allows for a black-box evaluation of zero-point vibrational corrections, completed in two successive steps, requiring a total of two molecular Hessians, 6K–11 molecular gradients, and 6K–11 property evaluations, K being the number of atoms. We apply the approach to the study of a number of electric and magnetic properties—the dipole and quadrupole moments, the static and frequency-dependent polarizability, the magnetizability, the rotational g tensor and the nuclear shieldings—of the molecules hydrogen fluoride, water, ammonia, and methane. Particular attention is paid to the importance of electron correlation and of the importance of the zero-point vibrational corrections for obtaining accurate estimates of molecular properties for a direct comparison with experiment.

The Acetylene−Ammonia Dimer as a Prototypical C−H···N Hydrogen-Bonded System: An Assessment of Theoretical Procedures

The effect of a variety of theoretical methods (HF, B3-LYP, MP2, QCISD, and CCSD(T)) and basis sets (from 6-31G(d) to 6-311+G(3df,2p)) on the calculated geometry and dimerization energy of the acetylene−ammonia dimer HCCH···NH3 is examined. The dimer has C3v symmetry with acetylene acting as the hydrogen bond donor. Our highest level calculations (viz. CCSD(T)/6-311+G(3df,2p) including BSSE correction) predict an equilibrium contact distance r(H···N) of 2.280 Å and an equilibrium binding energy ΔEe of 14.1 kJ mol-1. Incorporation of a scaled B3-LYP/6-311+G(3df,2p) zero-point vibrational correction leads to ΔE0 = 9.3 kJ mol-1. The less expensive CCSD(T)/6-311+G(3df,2p)//B3-LYP/6-311+G(3df,2p) procedure reproduces these benchmark energies and is therefore recommended for general application on small hydrogen-bonded systems. For larger hydrogen-bonded systems, the still less expensive B3-LYP/6-311+G(3df,2p)//B3-LYP/6-311+G(d,p) procedure is recommended, and this yields ΔE0 = 7.8 kJ mol-1 for the acetylene−ammonia dimer.

Calculation of static zero-point vibrational averaging corrections and other vibrational curvature contributions to polarizabilities and hyperpolarizabilities using field-induced coordinates

The zero-point static vibrational averaging (ZPVA) correction to the (hyper)polarizability is written in first-order as the sum of two contributions, one involving electric field derivatives and the other a sum over normal coordinate derivatives of the zero-point energy. It is shown that the sum over 3N−6 normal modes can be replaced by a single term using field-induced coordinates (FICs). A computational strategy that takes advantage of this simplification is presented and applied to a typical push–pull polyene NH2(CHCH)3NO2. From the dependence of the first-order ZPVA on the field-dependent equilibrium geometry, we also obtain other low-order static and dynamic vibrational curvature contributions to the (hyper)polarizabilities. The entire set of electronic and vibrational terms is partitioned into two different sequences, each of which exhibits rapid initial convergence for NH2(CHCH)3NO2 at the Hartree–Fock/6-31G+p level. Including electron correlation at the second-order Møller–Plesset (MP2) level, and the frequency dependence of the ZPVA correction is discussed. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 471–479, 2000

The correlation, relativistic, and vibrational contributions to the dipole moments, polarizabilities, and first and second hyperpolarizabilities of ZnS, CdS, and HgS

The dipole moments, dipole polarizabilities, and the first and second hyperpolarizabilities of the Group IIb sulfides have been calculated by using different high-level-correlated methods and including both the relativistic and vibrational contributions. The electron correlation effects have been studied at the levels of the second-order Møller–Plesset perturbation theory and the coupled-cluster methods. The relativistic contributions and the interference relativistic-correlation effects have been accounted for by using the spin-averaged Douglas–Kroll approximation. The vibrational properties (pure vibrational contributions and the zero-point vibrational averaging corrections) have been computed using CCSD(T) theory with and without relativistic corrections. The present pure electronic nonrelativistic results exhibit essentially the same pattern as that observed for similar molecules studied earlier. Most of the relativistic effects on dipole moments and dipole polarizabilities is accounted for at the level of the SCF approximation and rapidly increases with the nuclear charge of the heavy atom. The contribution of the relativistic-correlation interference terms has been found to be quite significant for axial components of the first and second dipole hyperpolarizabilities. All the properties reported here are static. This is the first study which reports on the relativistic contributions to hyperpolarizabilities as well as on vibrational effects upon both polarizabilities and hyperpolarizabilities of heavy metal (Group IIb) involving compounds. Thus the reported results add to the knowledge and understanding of the importance of the electron correlation, relativistic, and vibrational effects on electric properties of heavy molecules and extend the corresponding data beyond the linear response approximation. The reliability of the computed data is discussed in terms of the underlying approximations and limitations of methods used in this study.

Complete basis set, Gaussian, and hybrid density functional theory evaluation of the proton affinities of water and ammonia

Using Dunning’s basis set saturation approach, electronic energies for proton affinities for both ammonia and water were evaluated with hybrid density functional theory methods. It was demonstrated that the zero-point vibrational correction for both of these proton affinities computed at the B3LYP/6-311G(2d,2p) theory level is identical to the values obtained from experimental data and some other theoretical studies. Thus, computed proton affinities for ammonia and water are in excellent agreement with values obtained by G2 and CBSQ ab initio computational studies. The final proton affinity values are 201.8 kcal/mol for ammonia and 162.4 kcal/mol for water. These calculated values are 2–3 kcal/mol below the experimental values for ammonia and water, suggesting that these values might be too high.

Accurate structures and binding energies for small water clusters: The water trimer

The global minimum on the water trimer potential energy surface has been investigated by means of second-order Mo/ller-Plesset (MP2) perturbation theory employing the series of correlation-consistent basis sets aug-cc-pVXZ (X = D, T, Q, 5, 6), the largest of which contains 1329 basis functions. Definitive predictions are made for the binding energy and equilibrium structure, and improved values are presented for the harmonic vibrational frequencies. A value of 15.82±0.05 kcal mol−1 is advanced for the infinite basis set frozen core MP2 binding energy, obtained by extrapolation of MP2 correlation energies computed at the aug-cc-pVQZ MP2 geometry. Inclusion of core correlation, using the aug-cc-pCV5Z basis set, has been found to increase the binding energy by 0.08 kcal mol−1, and after consideration of core correlation and higher-order correlation effects, the classical binding energy for the water trimer is estimated to be 15.9±0.2 kcal mol−1. A zero-point vibrational correction of −5.43 kcal mol−1 has been computed from aug-cc-pVTZ MP2 harmonic vibrational frequencies. The accuracy of different computational schemes for obtaining the binding energies of the water dimer and trimer has been investigated, and computationally feasible methods are suggested for obtaining accurate structures and binding energies for larger water clusters.

An estimation of the isomerization energy of acetylene

An accurate prediction of the adiabatic energy difference separating acetylene and vinylidene is presented. The electronic energy contribution is estimated on the basis of CCSD(T) calculations using up to 400 contracted Gaussian basis functions together with extrapolation to the basis set limit; an additional correction is applied to account for residual electron correlation effects. Zero-point vibrational corrections are based on quartic force fields of both acetylene and vinylidene calculated at the CCSD(T) level. Also included are contributions for relativistic effects and the Born–Oppenheimer diagonal correction. The isomerization energy is estimated to be 15200±205 cm−1, which is somewhat lower than the best previous theoretical prediction.