The use of the mulliken approximation in bond-bond pair potentials describing rotational barriers

Possible sources of the error compensation effect of the Mulliken approximation for localized bond orbital overlap densities are discussed in terms of Ruedenberg's expansion. The Mulliken approximation has been applied previously for one- and two-electron integrals to simplify bond-bond pair potentials which describe barriers to internal rotation around single bonds.     

On bound states in two-body systems

The application of the Σ-separation method to the calculation of multicenter two-electron molecular integrals with Slater-type basis functions is reported. The approach is based on the approximation of a scalar component of the two-center atomic density by a two-center expansion over Slater-type functions. A least-squares fit was used to determine the coefficients of the expansion. The angular multipliers of the atomic density were treated exactly. It is shown that this approach can serve as a sufficiently accurate and fast algorithm for the calculation of multicenter two-electron molecular integrals with Slater-type basis functions. © 1995 John Wiley & Sons, Inc.     

Calculation of three-center nuclear attraction integral over Slater type orbitals in molecular coordinate system using Löwdin <Emphasis Type="Italic">α</Emphasis>-radial function and Guseinov’s two-center charge density expansion formulae

Using Löwdin α-radial function and the Guseinov’s charge density expansion formulae, the calculation of the three-center nuclear attraction integrals over Slater type orbitals in molecular coordinate system is performed. The proposed algorithm is especially useful for computation of multicenter-multielectron integrals that arise in the Hartree-Fock-Roothaan approximation, which plays a significant role for the study of electronic structure and electron-nuclei interaction properties of atoms, molecules and solids. The algorithm described in the present work is valid for the arbitrary values of quantum numbers, screening constants and internuclear distances. The calculation results are in good agreement with those obtained using the alternative evaluation procedure.     

Tensor decomposition in post-Hartree-Fock methods. I. Two-electron integrals and MP2

A new approximation for post-Hartree-Fock (HF) methods is presented applying tensor decomposition techniques in the canonical product tensor format. In this ansatz, multidimensional tensors like integrals or wavefunction parameters are processed as an expansion in one-dimensional representing vectors. This approach has the potential to decrease the computational effort and the storage requirements of conventional algorithms drastically while allowing for rigorous truncation and error estimation. For post-HF ab initio methods, for example, storage is reduced to O(d·R·n) with d being the number of dimensions of the full tensor, R being the expansion length (rank) of the tensor decomposition, and n being the number of entries in each dimension (i.e., the orbital index). If all tensors are expressed in the canonical format, the computational effort for any subsequent tensor contraction can be reduced to O(R(2)·n). We discuss details of the implementation, especially the decomposition of the two-electron integrals, the AO-MO transformation, the M?ller-Plesset perturbation theory (MP2) energy expression and the perspective for coupled cluster methods. An algorithm for rank reduction is presented that parallelizes trivially. For a set of representative examples, the scaling of the decomposition rank with system and basis set size is found to be O(N(1.8)) for the AO integrals, O(N(1.4)) for the MO integrals, and O(N(1.2)) for the MP2 t(2)-amplitudes (N denotes a measure of system size) if the upper bound of the error in the l(2)-norm is chosen as ε = 10(-2). This leads to an error in the MP2 energy in the order of mHartree.     

Modified all-valence INDO/spd method for ground and excited state properties of isolated molecules and molecular complexes

A new semiempirical all-valence method, GRINDOL (Ghost and Rydberg INDO ), based on the INDO approximation, is described. Linderberg–Seamans relation (extended to the d and Rydberg orbitals) for the resonance integrals and a new semitheoretical expression for the core-core repulsion term and energy correction including basis-set superposition error (intermolecular as well as intramolecular) has been applied. The proposed method enables calculation of ground and excited state properties. The ground state results (including intermolecular interactions) as well as the spectral properties are in reasonable agreement with relevant experimental (or ab initio) studies for isolated molecules, molecular complexes, and transition metal compounds. The method contains only one adjustable parameter, all two-center integrals and terms are only basis-set dependent. The one-center integrals are evaluated from the respective atomic terms.     

“Neglect of Diatomic Differential Overlap” in nonempirical quantum chemical orbital theories. IV. An examination of the justification of the neglect of diatomic differential overlap (NDDO) approximation

The Neglect of Diatomic Differential Overlap approximation is examined in terms of a polynomial expansion in Γ. The expansion is based upon the Legendre or Chebyshev approximation as developed in Part II. Analogous to the theorems of Chandler and Grader, NDDO cannot be justified for one-electron integrals and only partially for the two-electron repulsion integrals. © 1995 John Wiley & Sons, Inc.     

Iterative introduction of integral approximations with error bounds in SCF calculations

Another possible application of a previously reported approximation theory for electron repulsion integrals using rigorous error bounds is considered by incorporating the electron density matrix in the approximation scheme. Error bounds are set on the contribution of a given integral to the total energy for a given molecular wave-function; the wave-function is then refined cyclically and additional integrals are computed exactly if necessary until convergence is achieved.     

Connection between the regular approximation and the normalized elimination of the small component in relativistic quantum theory

The regular approximation to the normalized elimination of the small component (NESC) in the modified Dirac equation has been developed and presented in matrix form. The matrix form of the infinite-order regular approximation (IORA) expressions, obtained in [Filatov and Cremer, J. Chem. Phys. 118, 6741 (2003)] using the resolution of the identity, is the exact matrix representation and corresponds to the zeroth-order regular approximation to NESC (NESC-ZORA). Because IORA (=NESC-ZORA) is a variationally stable method, it was used as a suitable starting point for the development of the second-order regular approximation to NESC (NESC-SORA). As shown for hydrogenlike ions, NESC-SORA energies are closer to the exact Dirac energies than the energies from the fifth-order Douglas-Kroll approximation, which is much more computationally demanding than NESC-SORA. For the application of IORA (=NESC-ZORA) and NESC-SORA to many-electron systems, the number of the two-electron integrals that need to be evaluated (identical to the number of the two-electron integrals of a full Dirac-Hartree-Fock calculation) was drastically reduced by using the resolution of the identity technique. An approximation was derived, which requires only the two-electron integrals of a nonrelativistic calculation. The accuracy of this approach was demonstrated for heliumlike ions. The total energy based on the approximate integrals deviates from the energy calculated with the exact integrals by less than 5 x 10(-9) hartree units. NESC-ZORA and NESC-SORA can easily be implemented in any nonrelativistic quantum chemical program. Their application is comparable in cost with that of nonrelativistic methods. The methods can be run with density functional theory and any wave function method. NESC-SORA has the advantage that it does not imply a picture change.     

Simplified ab-initio calculations for molecular systems

A method is described for reducing a large part of the arithmetic of exact ab-initio SCF molecularorbital calculations based on Slater-type-orbitals without noticeable loss of numerical accuracy. The procedure involves the transformation to Löwdin orthogonalized orbitals and then invoking the NDDO approximation. The three- and four-centre two-electron integrals required are estimated by a truncated Ruedenberg expansion. All one-electron integrals are evaluated exactly. No empirical parameters are employed. Numerical tests on CO, OF₂, O₃ and ONF show that the NDDO approximation is very accurate for Löwdin functions and that the Ruedenberg expansion is arithmetically satisfactory for the SCF MO calculations.     

Three-center expansion of electron repulsion integrals with linear combination of atomic electron distributions

A new systematic way of constructing auxiliary basis functions for approximating the evaluation of electron repulsion integrals is proposed and applied to SCF and MCSCF wavefunction calculations. In the approximation, the one-electron density is expanded in terms of a linear combination of atomic electron distributions (LCAD), and the four-center two-electron repulsion integrals are reduced to the three- and two-center quantities. This results in a high-accuracy approximation as well as a large reduction in disk storage and input/output requirement, proportional to N³ rather than N⁴, N being the number of basis functions. Numerical results indicate that the error from the present approximation decreases as the size of molecular basis functions increases and that the LCAD version of MCSCF calculations requires only a fractional amount of the CPU time required in the conventional procedure without loss of accuracy.     

Efficient evaluation of one-center three-electron Gaussian integrals

 An algorithm is presented for the efficient evaluation of two types of one-center three-electron Gaussian integrals. These integrals are required to avoid the resolution-of-identity (RI) approximation in explicitly correlated linear R12 methods. Without the RI approximation, it is possible to enforce rigorously the strong orthogonality of the second-order M?ller–Plesset R12 ansatz. A test calculation is performed using atomic Gaussian-type orbitals of the neon atom. Received: 21 November 2000 / Accepted: 6 April 2001 / Published online: 9 August 2001     

Evaluation of expansion coefficients for translation of Slater‐type orbitals using complete orthonormal sets of exponential‐type functions

By the use of exponential‐type functions (ETFs) the simpler formulas for the expansion of Slater‐type orbitals (STOs) in terms of STOs at a displaced center are derived. The expansion coefficients for translation of STOs are presented by a linear combination of overlap integrals. The final results are of a simple structure and are, therefore, especially useful for machine computations of arbitrary multielectron multicenter molecular integrals over STOs that arise in the Hartree–Fock–Roothaan approximation and also in the Hylleraas correlated wave function method for the determination of arbitrary multielectron properties of atoms and molecules. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 126–129, 2001     

Molecular integrals over the gauge-including atomic orbitals. II. The Breit-Pauli interaction

Each accompanying coordinate expansion (ACE) formula is derived for each of the orbit-orbit interaction, the spin-orbit coupling, the spin-spin coupling, and the contact interaction integrals over the gauge-including atomic orbitals (GIAOs) by the use of the solid harmonic gradient (SHG) operator. Each ACE formula is the general formula derived at the first time for each of the above molecular integrals over GIAOs. These molecular integrals are arising in the Breit-Pauli two-electron interaction for a relativistic calculation. We may conclude that we can derive a certain ACE formula for any kind of molecular integral over solid harmonic Gaussian-type orbitals by using the SHG operator. The present ACE formulas will be useful, for example, for a calculation of a molecule in a uniform magnetic field, for a relativistic calculation, and so on, with the GIAO as a basis function.     

Ab initio molecular orbital calculations on large lattice cluster models: Use of translational symmetry

Summary Translational symmetry has been shown to be useful in the calculation of electronic structures of large lattice models. The number of unique integrals has been derived for cases of different dimensionality. For the unique integrals zero screening and approximation methods are described. The method has been applied to arrays of hydrogen atoms and to a zincblende surface model. When the size of the system is increased the translationally unique integrals are shown to become either zero or they can be calculated by simple coulombic approximations.     

Large scale polarizability calculations using the approximate coupled cluster model CC2 and MP2 combined with the resolution-of-the-identity approximation

We present an implementation of static and frequency-dependent polarizabilities for the approximate coupled cluster singles and doubles model CC2 and static polarizabilities for second-order Mo?ller-Plesset perturbation theory. Both are combined with the resolution-of-the-identity approximation for electron repulsion integrals to achieve unprecedented low operation counts, input-output, and disc space demands. To avoid the storage of double excitation amplitudes during the calculation of derivatives of density matrices, we employ in addition a numerical Laplace transformation for orbital energy denominators. It is shown that the error introduced by this approximation is negligible already with a small number of sampling points. Thereby an implementation of second-order one-particle properties is realized, which avoids completely the storage of quantities scaling with the fourth power of the system size. The implementation is tested on a set of organic molecules including large fused aromatic ring systems and the C(60) fullerene. It is demonstrated that exploiting symmetry and shared memory parallelization, second-order properties for such systems can be evaluated at the CC2 and MP2 level within a few hours of calculation time. As large scale applications, we present results for the 7-, 9-, and 11-ring helicenes.     

An analysis of the zero differential overlap approximation. Towards an improved semiempirical MO method beyond it

Summary Some systematic errors of the zero differential overlap (ZDO) approximation in semiempirical molecular orbital (MO) methods are discussed. In electron methods, a power series expansion of the inverse square rootS ^–1/2 of the overlap matrix and application of the Mulliken approximation to the two-electron integrals show that the ZDO Hamiltonian coincides with the Hamiltonian obtained by explicit performance of the Löwdin transformation up to first-order terms of diatomic overlap densities. Higher than first-order terms lead to a systematic up-shift of the canonical MO energies. Although a power series expansion ofS ^–1/2 is no longer possible in all-valence-electron methods, the MO levels resulting from the ZDO approximation are also systematically placed at too low energies, especially the low-lying occupied and the virtual MOs. A method based on explicit performance of the Löwdin transformation and retaining the simplicity of the ZDO approach for the calculation of Fock matrix elements is developed. The parameters of this method are obtained by very simple manipulations of the original ZDO parameters. Numerical calculations show that a considerable improvement of the MO energy spectrum in the inner valence region can be obtained in this way     

A semiempirical model for the two-center repulsion integrals in the NDDO approximation

A self-consistent formalism is proposed for the two-center electron repulsion integrals in the NDDO approximation, based on their expansion in terms of multipole-multipole interactions and free from adjustable parameters.     

Relativistic calculation of nuclear magnetic shielding tensor using the regular approximation to the normalized elimination of the small component. II. Consideration of perturbations in the metric operator

A previous relativistic shielding calculation theory based on the regular approximation to the normalized elimination of the small component approach is improved by the inclusion of the magnetic interaction term contained in the metric operator. In order to consider effects of the metric perturbation, the self-consistent perturbation theory is used for the case of perturbation-dependent overlap integrals. The calculation results show that the second-order regular approximation results obtained for the isotropic shielding constants of halogen nuclei are well improved by the inclusion of the metric perturbation to reproduce the fully relativistic four-component Dirac-Hartree-Fock results. However, it is shown that the metric perturbation hardly or does not affect the anisotropy of the halogen shielding tensors and the proton magnetic shieldings.     

Random-phase-approximation calculations on triplet spectra of conjugated molecules

Random-phase approximations (RPA) have been applied to the calculation of the triplet π-π* transition spectra of 18 conjugated molecules in the framework of Pariser-Parr-Pople approximations. It is found that the normal RPA (n-RPA) shows the triplet instability for most molecules in the Nishimoto-Mataga approximation of electron-repulsion integrals. However, it is shown that this instability can be circumvented by the use of the renormalized RPA (r-RPA) in which the correlated ground states are calculated by the second-order perturbation theory. It is also shown that even in the n-RPA the suitable parametrization of electron-repulsion integrals removes this instability. It is ascertained that such an increasing order of energies as ω(n-RPA)<ω(Tamm-Dancoff approximation)<ω(r-RPA) holds for most of energy levels.