Core–Valence correlations and relativistic effects in heavy atoms for molecular applications

Theoretical core effective potential methods are widely used in valence-only electron molecular calculations. These methods, which imply the frozen-core approximation, work well for the elements of the righthand side of the periodic table but are often unrealistic for metallic elements with highly polarizable cores. For these atoms one has to consider the polarization of the cores under the influence of the electric field created by the valence electrons. Moreover, relativistic corrections must be added for heavy atoms. Various theoretical approaches of core–valence interactions (polarization and core–valence correlations) will be reviewed, with a special emphasis on practical methods of calculation. The problem of handling the relativistic effects will mainly be discussed within the two-component Pauli formalism. It will be shown that the Foldy–Wouthuysen transformation is not the unique way for deriving relativistic corrections and that the second-order Dirac equation also provides a good starting point for obtaining relativistic corrections. Analytical exact results are given for the hydrogen atom. The accuracy of this approach is tested on many-electron atoms and molecules. It is finally shown that the problem of the core-valence separation is relevant to the general methodology of effective Hamiltonians that seems to provide the best promising way for filling the gap between the semiempirical and purely theoretical ab initio methods.     

Quasi-relativistic approximation in the SCF-MO-LCAO method

A quasi-relativistic approach to the MO-LCAO method is formulated taking into account the relativistic effects with an accuracy up to (v/c)² terms, the relativistic part of the electronic interaction in the Hamiltonian being neglected. In the framework of this approximation a set of SCF equations of the Roothaan form is derived; here only the relativistic analogue to the closed shell systems with one-determinant wave functions is considered. In so doing three types of relativistic corrections arise which are quite similar to those of the Pauli equation for one-electron atoms. The new matrix elements appearing due to these corrections can be reduced to some common integrals, which have to be calculated with relativistic radial atomic functions. The method allows a semi-empirical approach to the problem and does not require the Dirac four-component atomic functions (unknown in the most cases), thus making possible approximate quasi-relativistic electronic structure calculations of heavy-atom compounds.     

Generalized spherical functions in the theory of many-electron atoms

A new method for finding non-relativistic and relativistic wave-functions of an electron moving in the field of a nuclear charge in the jj coupling scheme is proposed. It is based on the usage of generalized spherical functions. The mathematical apparatus necessary to find the expressions for matrix elements of the non-relativistic and relativistic energy or electron transition operators is developed. The formulas obtained for these matrix elements are more convenient than those usually used in jj coupling scheme; only their radial integrals and some phase multipliers depend on orbital quantum numbers.     

Singular and nonsingular three-body integrals for exponential wave functions

Integrals which are individually singular, but which may be combined to yield convergent expressions, are needed for computations of relativistic effects and various properties of atomic and quasiatomic systems. As computations become more detailed and precise, more such integrals are required. This paper presents general formulas for the radial parts of the singular and nonsingular (regular) integrals that occur when three-body systems are described using wave functions that include exponentials in all three interparticle coordinates. Our results are compared with those found in the literature for some of the integrals, and are also shown to be consistent with previously reported results for Hylleraas functions (a limiting case in which one of the exponential parameters is set to zero).     

Improved partition–expansion of two‐center distributions involving slater functions

The calculation of the electronic structure of large systems is facilitated by the substitution of the two‐center distributions by their projections on auxiliary basis sets of one‐center functions. An alternative is the partition–expansion method in which one first decides what part of the distribution is assigned to each center, and next expands each part in spherical harmonics times radial factors. The method is exact, requires neither auxiliary basis sets nor projections, and can be applied to Gaussian and Slater basis sets. Two improvements in the partition–expansion method for Slater functions are reported: general expressions valid for arbitrary quantum numbers are derived and the efficiency of the procedure is increased giving analytical solutions to integrals previously computed by numerical quadrature. The efficiency of the new version is assessed in several molecules and the advantages over the projection methods are pointed out. © 2013 Wiley Periodicals, Inc.     

Relativistic theory for indirect nuclear spin–spin couplings within the polarization propagator approach

We present a relativistic theory for the nuclear spin–spin coupling tensor within the polarization propagator approach using the particle-hole Dirac–Coulomb–Breit Hamiltonian and the full four-component wave function. We give explicit expressions for the coupling tensor in the random-phase approximation, neglecting the Breit interaction. A purely relativistic perturbative electron–nuclear Hamiltonian is used and it is shown how the single relativistic contribution to the coupling tensor reduces to Ramsey's three second-order terms (Fermi contact, spin–dipole, and paramagnetic spin–orbit) in the nonrelativistic limit. The principal propagator becomes complex and the leading property integrals mix atomic orbitals of different parity. The well-known propagator expressions for the coupling tensor in the nonrelativistic limit is obtained neglecting terms of the order c^?n (n ? 1). © 1993 John Wiley & Sons, Inc.     

Molecular integrals for Gaussian and exponential‐type functions: Shift operators

Basis functions with arbitrary quantum numbers can be attained from those with the lowest numbers by applying shift operators. We derive the general expressions and the recurrence relations of these operators for Cartesian basis sets with Gaussian and exponential radial factors. In correspondence, the expressions of molecular integrals involving functions with arbitrary quantum numbers can be obtained by applying these operators on the integrals with the lowest quantum numbers. Since the original form of the shift operators is not appropriate to deal with integrals, we give their representation in terms of derivatives with respect to the parameters on which these integrals explicitly depend. Moreover, we translate the recurrence relations to the new representation and, finally, we analyze the general expressions ot the molecular integrals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 137–145, 2000     

New angle‐dependent potential energy function for backbone–backbone hydrogen bond in protein–protein interactions

Backbone–backbone hydrogen bonds (BBHBs) are one of the most abundant interactions at the interface of protein–protein complex. Here, we propose an angle‐dependent potential energy function for BBHB based on density functional theory (DFT) calculations and the operation of a genetic algorithm to find the optimal parameters in the potential energy function. The angular part of the energy funtion is assumed to be the product of the power series of sine and cosine functions with respect to the two angles associated with BBHB. Two radial functions are taken into account in this study: Morse and Leonard‐Jones 12‐10 potential functions. Of these two functions under consideration, the former is found to be more accurate than the latter in terms of predicting the binding energies obtained from DFT calculations. The new HB potential function also compares well with the knowledge‐based potential derived by applying Boltzmann statistics for a variety of protein–protein complexes in protein data bank. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Transformed Dirac equation for the hydrogen atom,comparison with previous approaches in momentum space,and the anomalous Zeeman effect in momentum representation

The solution of a unitarily transformed Dirac equation for the hydrogenic electron in zero magnetic field is investigated here. The momentum‐space representation is adopted as a natural recourse. The spinor part of the transformed wavefunction in momentum space can be easily prescribed for a central potential. Hence, for the Coulomb potential, a pair of equations is obtained for the radial components in momentum space. It is shown that starting from these radial equations, one can recover the equations previously derived by Rubinowicz, Lévy, and Lombardi for the problem of the Dirac hydrogen atom in momentum space. This establishes equivalence among different approaches based on the momentum representation, including the current treatment. The recovery of the equations due to Rubinowicz permits the exact eigenvalues to be written down and exact expressions to be derived for the radial components of the transformed wavefunction in momentum space. A new approach is adopted to carry out a reduction to the nonrelativistic regime and the nonrelativistic limit. At first the transformed momentum‐space equation for the hydrogen atom is rewritten in terms of the hyperspherical coordinates. The zeroth‐order solutions of the new equation are recovered in the limit c → ∞ where c is the speed of light. These are manifestly separable into positive‐ and negative‐energy forms. For positive energy, these solutions have nonvanishing upper components that are two‐component spinors. The latter exactly correspond to the single‐component, nonrelativistic, momentum‐space solutions derived by Fock. It is shown that when the upper component is corrected through first order in v²/c² but the separability is still maintained for the transformed wavefunction, one retrieves the Pauli equation in momentum space. It is also shown that for a hydrogen atom placed in a uniform magnetic field, the nonvanishing momentum‐space matrix elements representing the anomalous Zeeman effect have a simple form, namely, the product of a radial integral and an angular integral. These integrals are equal to the well‐known radial and angular integrals in coordinate representation. The matrix elements can be easily evaluated. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Pair-excitation MCSCF treatment of small molecules in an optimized Slater–Transform–Preuss basis set

Pair-excitation multiconfigurational self-consistent field (PEMCSCF ) treatment of 11 small molecules (LiH, BeH₂, BH₃, BF, CH₄, C₂H₄, C₂H₂, CH₂O, NH₃, H₂O, and HF) has been carried out in a minimum basis set of Slater Transform Preuss functions as fitted by six cartesian gaussians (STP -6G). The advantages of accuracy without using a split basis are shown by comparison to familiar 4-31G and 6-31G calculations using molecular geometries optimized with STO -6G basis sets. A benefit is shown for the use of minimum basis fitted to STP functions: they overemphasize long-tail radial dependence to achieve long range basis sensitivity without increasing the basis size at the AO -to-MO transformation step in the configuration interaction portion of the MCSCF algorithm. Fully optimized STP -6G parameters are given and appear to be transferable as shown for acrolein. A FORTRAN listing of the full least squares fitting algorithm is available* for in situ generation of STP -6G orbitals energetically superior to 4-31G, or a less accurate STP -6G 1S, 2S, and 2P basis may be scaled directly as if they were STO -6G functions, but with considerably lower energy that with an STO -6G basis.     

On the long way from the Coulombic Hamiltonian to the intermolecular energy surfaces: Concluding remarks at the tromsø conference,June 20–22, 1989

After a brief review of the main results given at the conference, the general properties of the Coulombic Hamiltonian for a system of electrons moving in a framework of moving atomic nuclei—considered as point charges—are discussed. Since this Hamiltonian is invariant under translations and rotations, the total momentum and the total angular momentum are constants of motion, which means that it is possible to separate the motion of the center of mass and the rotation of the system as a whole. Even if these separations are simple in principle, they lead to a mixing of the electronic and nuclear coordinates that complicates the transformed Hamiltonian. The general features of this Hamiltonian are discussed both in pure quantum mechanics and general quantum theory dealing with wave functions Ψ respective density matrices ρ or system operators T. The principles of the latter are derived from five simple axioms, and it is shown that pure quantum mechanics is a special case of the general theory and that the analogy between these two approaches is essential for the “economy of thinking.” It is indicated that the general theory of the shape and topology of the energy surface 〈H〉 = TrHΓ and its critical points, as a function of the system operator Γ involving both electronic and nuclear coordinates, is a very difficult mathematical problem and that calculation of this surface even for simple molecular systems represents a formidable computational problem, which has to be solved in order to be able to understand the nature of chemical reactions from first principles.     

Electronic structure of molecules using one-component wave functions and relativistic effective core potentials

A one-component approach to molecular electronic structure is discussed that includes the dominant relativistic effects on valence electrons and yet allows the use of the traditional quantum-chemistry techniques. The approach starts with one-component Cowan–Griffin relativistic orbitals that successfully incorporate the effects of the mass-velocity and Darwin terms present in more complicated wave functions such as the Dirac–Hartree–Fock. The approach then constructs “relativistic” effective core potentials (RECPS ) from these orbitals, and uses these to bring the relativistic effects into the molecular electronic calculations. The use of effective one-electron spin-orbit operators in conjunction with these one-component wave functions to include the effects of spin-orbit coupling is discussed. Applications to molecular systems involving heavy atoms and comparisons with available spectroscopic data on molecular geometries and excitation energies are presented. Finally, a new approach to the construction of RECPS encompassing the Hamiltonian and shapeconsistent approach is presented together with a novel analysis of the long-range behavior of the RECPS .     

The sound velocities in dense fluids from distribution functions

The sound velocities and adiabatic compressibilities in dense fluids have been evaluated using three known analytical expressions for radial distribution functions (RDFs). Using such approach not only tests the power of distribution functions theory in predicting the sound velocities and adiabatic compressibilities, but also specifies better expressions in determining these properties. To calculate these quantities, the variation of RDF with density and temperature is required. Therefore, we should have analytical expressions which explicitly present RDF as a function of temperature, density and interparticle distance. It is shown that if an expression is used which properly presents RDFs as a function of interparticle distance, density and temperature, it is possible to calculate sound velocities and adiabatic compressibilities from distribution function theory.     

Confinement of hydrogen atom with Dirac equation

The problem of a particle confined in a spherical cavity is studied with the Dirac equation. A hard confinement is obtained by forcing the large component to vanish at the cavity radius. It is shown that the small component cannot vanish simultaneously at this radius. In the case of a confined hydrogen atom, the energies are given by an implicit equation. For some values of the radius, explicit analytical expressions of the energy exist like in the nonrelativistic case. Very accurate energies and wave functions are obtained with the Lagrange-mesh method with few mesh points. To this end, two differently regularized Lagrange-Jacobi bases associated with the same mesh are used for the large and small components. The importance of relativistic effects is discussed for hydrogen-like ions. The validity of this definition of hard confinement is discussed with a soft-confinement model studied with the R-matrix method.     

A new treatment of the vibration–Rotation eigenvalue problem for a diatomic molecule

The problem of the determination of the vibration–rotation eigenvalue in diatomic molecules is considered. An eigenvalue equation totally independent from the eigenfunction is written for any potential, analytical or numerical. This equation uses uniquely the vibration–rotation canonical functions; its resolution is reduced to that of a simple and classical numerical problem. Examples of numerical applications for analytical (Morse) and numerical potentials are presented. It is shown that the vibrational eigenvalues deduced from the eigenvalue equation are within 10^–6 cm^–1 of the exact values. Comparison with conventional methods are presented and discussed.     

Optimized Gaussian basis sets for use with relativistic effective (core) potentials: K,Ca, Ga—Kr

Correlation-consistent valence basis sets were developed for the third-row main block elements (K, Ca, Ga—Kr) for use with relativistic effective core potentials. These basis sets are somewhat larger than double-zeta in size, with polarization functions, and are balanced for use in both Hartree–Fock and correlation calculations. Spin–orbit splittings for atoms and molecules are calculated and compared to experiment. These calculations use the approximate spin–orbit operator from the relativistic effective core potentials. The use of these results in the calculation of accurate thermochemical data is discussed. © 1997 John Wiley & Sons, Inc.

1 This article is a US Government work and, as such, is in the public domain in the United States of America.