Variational solution of the dirac equation within a multicentre basis set of gaussian functions

The Dirac equation for one-electron polynuclear systems is solved variationally within an analytical multicentre basis set. Each of the four components of the molecular spinor is expanded separately into a scalar basis set. The expansion coefficients are obtained from the variational principle. Calculations using gaussian functions are performed for hydrogen-like atoms and for hydrogen-ion-like molecules with constituents of nuclear charge Z = 1, 50 and 90.     

Exact solution of the adsorption integral equation for a heterogeneous surface

An exact method of solution of the adsorption integral equation using finite and realistic limits of the heats of adsorption and which is applicable to any analytic site-energy distribution is given. The optical feature is the postulation and use of a set of transformations to reduce the finite-limit problem to one of limits (0, ∞). The transformed integral equation is then inverted using the theory of Stieltjes transforms. The method thus avoids the limitations inherent in earlier work which employed restrictive assumptions about the heats of adsorption.     

Approximate relativistic Hartree-Fock equations and their solution within a minimum basis set of slater-type functions

Relativistic closed-shell atoms are treated by the use of a specific approximation for the small component of the one-electron Dirac spinors. It is assumed that the large and the small component are interconnected by a parameter-dependent relation which is formally analogous to that of the one-electron system. Subject to this constraint, the total energy is varied with respect to the large components. The resulting eigenvalue equations for the large components contain only regular potential terms and reduce to the familiar Hartree-Fock equations in the limit of infinite velocity of light. Analytical solution of these approximate relativistic Hartree-Fock equations is achieved using a minimum basis set of Slater-type functions for the expansion of the radial part of the large components. Total relativistic energies, orbital energies, orbital exponents and mean radii are calculated for the ground states of He, Be, Ne, Mg, Ar, Kr Xe and Cu⁺. Dedicated to Prof. O. E. Polansky on occasion of his 60th birthday.     

On the gauge invariance of the schrodinger equation

The problem of gauge invariance in relation to (approximate) calculations of molecular properties is considered. We show that these difficulties are not avoided by making a unitary transformation to a formalism which is independent of the electromagnetic field potentials.     

Solving the low dimensional Smoluchowski equation with a singular value basis set

Reaction kinetics on free energy surfaces with small activation barriers can be computed directly with the Smoluchowski equation. The procedure is computationally expensive even in a few dimensions. We present a propagation method that considerably reduces computational time for a particular class of problems: when the free energy surface suddenly switches by a small amount, and the probability distribution relaxes to a new equilibrium value. This case describes relaxation experiments. To achieve efficient solution, we expand the density matrix in a basis set obtained by singular value decomposition of equilibrium density matrices. Grid size during propagation is reduced from (100–1000)^N to (2–4)^N in N dimensions. Although the scaling with N is not improved, the smaller basis set nonetheless yields a significant speed up for low‐dimensional calculations. To demonstrate the practicality of our method, we couple Smoluchowsi dynamics with a genetic algorithm to search for free energy surfaces compatible with the multiprobe thermodynamics and temperature jump experiment reported for the protein α₃D. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Self-consistent solution of the Dyson equation for atoms and molecules within a conserving approximation

We have calculated the self-consistent Green's function for a number of atoms and diatomic molecules. This Green's function is obtained from a conserving self-energy approximation, which implies that the observables calculated from the Green's functions agree with the macroscopic conservation laws for particle number, momentum, and energy. As a further consequence, the kinetic and potential energies agree with the virial theorem, and the many possible methods for calculating the total energy all give the same result. In these calculations we use the finite temperature formalism and calculate the Green's function on the imaginary time axis. This allows for a simple extension to nonequilibrium systems. We have compared the energies from self-consistent Green's functions to those of nonselfconsistent schemes and also calculated ionization potentials from the Green's functions by using the extended Koopmans' theorem.     

Diffusion-reaction approach to electronic relaxation in solution. Exact solution for delta function sink models

We give a general method for finding the exact solution for the problem of electronic relaxation in solution, modelled by a particle undergoing diffusive motion in a potential in presence of a delta function sink. The diffusive motion is described by the Smoluchowski equation and the sink could be a delta function of arbitrary position and strength. The solution requires the knowledge of the Laplace transform of the Green’s function for the motion in the absence of the sink. We use the method to find the solution of the problem in the case where the diffusive motion is on a parabolic potential. This has been an unsolved problem for some time and is of considerable importance as a model for non-radiative electronic relaxation of a molecule in solution. The solution is analyzed to obtain the viscosity and temperature dependences of the rate constants.     

Extrapolation to the limit of a complete basis set for electronic structure calculations on the N2 molecule

Results obtained from nonrelativistic electronic structure calculations using finite Gaussian basis sets are extrapolated to the limit of a complete basis set, employing the results of explicitly correlated coupled-cluster calculations including singles and doubles substitutions (CCSD). For N₂, the basis-set limits for the electronic binding energy, equilibrium bond length and harmonic vibrational wave number are established for the CCSD model including a perturbative correction for triples substitutions and for the internally contracted multireference configuration interaction method. The resulting numbers are in good agreement with experimental values. Received: 2 December 1997 / Accepted: 3 February 1998 / Published online: 17 June 1998     

Formulation of the LCAS MS SCF method within the Gaussian basis set

Within the presented LCAS MS (linear combination of atomic spinors–molecular spinors) SCF formalism both large and small components of the spinor radial parts have been expanded within the Gaussian basis set. The respective expressions for matrix elements as well as for one- and two-electron integrals are given.     

The sensitivity of B3LYP atomization energies to the basis set and a comparison of basis set requirements for CCSD(T) and B3LYP

The atomization energies of the 55 G2 molecules are computed using the B3LYP approach with a variety of basis sets. The 6–311 + G(3df) basis set is found to yield superior results to those obtained using the augumented-correlation-consistent valence-polarized triple-zeta set. The atomization energy of SO₂ is found to be the most sensitive to basis set and is studied in detail. Including tight d functions is found to be important for obtaining good atomization energies. The results for SO₂ are compared with those obtained using the coupled-cluster singles and doubles approach including a perturbational estimate of the triple excitations.     

A comparison of the behavior of functional/basis set combinations for hydrogen-bonding in the water dimer with emphasis on basis set superposition error

We evaluate the performance of ten functionals (B3LYP, M05, M05-2X, M06, M06-2X, B2PLYP, B2PLYPD, X3LYP, B97D, and MPWB1K) in combination with 16 basis sets ranging in complexity from 6-31G(d) to aug-cc-pV5Z for the calculation of the H-bonded water dimer with the goal of defining which combinations of functionals and basis sets provide a combination of economy and accuracy for H-bonded systems. We have compared the results to the best non-density functional theory (non-DFT) molecular orbital (MO) calculations and to experimental results. Several of the smaller basis sets lead to qualitatively incorrect geometries when optimized on a normal potential energy surface (PES). This problem disappears when the optimization is performed on a counterpoise (CP) corrected PES. The calculated interaction energies (ΔEs) with the largest basis sets vary from -4.42 (B97D) to -5.19 (B2PLYPD) kcal/mol for the different functionals. Small basis sets generally predict stronger interactions than the large ones. We found that, because of error compensation, the smaller basis sets gave the best results (in comparison to experimental and high-level non-DFT MO calculations) when combined with a functional that predicts a weak interaction with the largest basis set. As many applications are complex systems and require economical calculations, we suggest the following functional/basis set combinations in order of increasing complexity and cost: (1) D95(d,p) with B3LYP, B97D, M06, or MPWB1k; (2) 6-311G(d,p) with B3LYP; (3) D95++(d,p) with B3LYP, B97D, or MPWB1K; (4) 6-311++G(d,p) with B3LYP or B97D; and (5) aug-cc-pVDZ with M05-2X, M06-2X, or X3LYP.     

On the effectiveness of CCSD(T) complete basis set extrapolations for atomization energies

The leading cause of error in standard coupled cluster theory calculations of thermodynamic properties such as atomization energies and heats of formation originates with the truncation of the one-particle basis set expansion. Unfortunately, the use of finite basis sets is currently a computational necessity. Even with basis sets of quadruple zeta quality, errors can easily exceed 8 kcal/mol in small molecules, rendering the results of little practical use. Attempts to address this serious problem have led to a wide variety of proposals for simple complete basis set extrapolation formulas that exploit the regularity in the correlation consistent sequence of basis sets. This study explores the effectiveness of six formulas for reproducing the complete basis set limit. The W4 approach was also examined, although in lesser detail. Reference atomization energies were obtained from standard coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) calculations involving basis sets of 6ζ or better quality for a collection of 141 molecules. In addition, a subset of 51 atomization energies was treated with explicitly correlated CCSD(T)-F12b calculations and very large basis sets. Of the formulas considered, all proved reliable at reducing the one-particle expansion error. Even the least effective formulas cut the error in the raw values by more than half, a feat requiring a much larger basis set without the aid of extrapolation. The most effective formulas cut the mean absolute deviation by a further factor of two. Careful examination of the complete body of statistics failed to reveal a single choice that out performed the others for all basis set combinations and all classes of molecules.     

Full Cl extrapolation compared to explicit full Cl for H2O in a double-zeta basis

Successive mMain subspace MRD Cl variational energies are shown to form the basis of an extrapolation of high precision to the corresponding exact full Cl energy for H₂O in a double-zeta basis. While the highest mMain subspaces considered (m = 1, 3, 6, 11, 18, 26, 35 and 45) involved only ≈5% of the full Cl space of 256473 SAFs, the error due to extrapolation to the full Cl result is only 0.21 kJ mol^?1, comparable to the precision to which we compute the Cl energies (≈0.05 kJ mol^?1).     

A method for the calculation of transition moments between electronic states of molecules using a different one-electron basis set for each state

A state-specific approach to the calculation of transition moments between molecular electronic states requires that the wavefunction for each state is expanded in its optimum one-electron basis and that nonorthonormal basis techniques are used for the calculation of the transition moment integrals. A method has been developed for carrying out such nonorthonormal basis calculations, based on the corresponding orbitals transformation and appropriately defined density matrices, which may be used with configuration interaction (CI ) wavefunctions. Further improvements of the method have resulted in a decrease in the time required for the calculations and thus allow its application with moderately large CI expansions for each state. Nonorthonormal basis calculations on transition moments in H₂O have been carried out using the above method. The results are in agreement with those of large MRD -CI calculations.     

Estimating the basis set superposition error in the CCSD(T)(F12) explicitly correlated method using the example of a water dimer

Changes in the basis set superposition errors upon transitioning from conventional CCSD(T) to the CCSD(T)(F12) explicitly correlated method is studied using the example of a water dimer. A comparison of the compensation errors for CCSD(T) and CCSD(T)(F12) reveals a substantial reduction in the superposition error upon use of the latter. Numerical experiments with water dimers show it is possible theoretically predict an equilibrium distance between oxygen atoms that is similar to the experimental data (2.946 Å), as is the predicted energy of dissociation of a dimer (5.4 ± 0.7 kcal/mol). It is found that the structural and energy parameters of hydrogen bonds in water dimers can be calculated precisely even with two-exponential correlation-consistent basis sets if we use the explicitly correlated approach and subsequently correct the basis set superposition error.     

Basis set approach to solution of poisson equation for small molecules immersed in solvent

The problem of how to calculate the electrostatic interactions between molecules and a solvent is a very important one in theoretical chemistry and biophysics. One of the more commonly used methods has been to represent the solvent by a dielectric continuum and then to solve the Poisson (or the Poisson-Boltzmann) equation for the potential due to the charge distribution of the solute. The solution of the equation has, up to now, been largely carried out using finite-difference grid-based methods. In this article, we investigate the use of an alternative method, based on a basis set expansion of the potential. The choice of basis functions, the representation of the dielectric function and the accuracy that can be obtained are discussed and illustrated by example calculations on small molecules. © 1997 by John Wiley & Sons, Inc.     

Ab initio calculations,using a small Gaussian basis set,of the electronic structure of the sulphate ion

Ab initio molecular orbital calculations of the electronic structure of the sulphate ion have been performed in which three Gaussian-type functions are used to simulate each member of a minimal basis of Slater-type orbitals. Comparative calculations on H₂S show that such a basis excellently reproduces the properties of the valence electrons given by calculations in a Slater basis. The expansion of the basis by the addition of sulphur 3d orbitals results in a large decrease in the molecular energy (1 a.u.) and has a pronounced effect on the ordering and energy of the molecular orbitals. The results of a number of semiempirical schemes are discussed in the light of these results.