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.     

Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. I. Three-electron integrals

A method to evaluate the nonrelativistic electron-repulsion, nuclear attraction and kinetic energy three-electron integrals over Slater orbitals appearing in Hylleraas-CI (Hy-CI) electron structure calculations on atoms is shown. It consists on the direct integration over the interelectronic coordinate r _ij and the sucessive integration over the coordinates of one of the electrons. All the integrals are expressed as linear combinations of basic two-electron integrals. These last are solved in terms of auxiliary two-electron integrals which are easy to compute and have high accuracy. The use of auxiliary three-electron ones is avoided, with great saving of storage memory. Therefore this method can be used for Hy-CI calculations on atoms with number of electrons N ≥ 5. It has been possible to calculate the kinetic energy also in terms of basic two-electron integrals by using the Hamiltonian in Hylleraas coordinates, for this purpose some mathematical aspects like derivatives of the spherical harmonics with respect to the polar angles and recursion relations are treated and some new relations are given.     

On the use of spatial symmetry in atomic-integral calculations: An efficient permutational approach

The minimal number of independent nonzero atomic integrals that occur over arbitrarily oriented basis orbitals of the form ?(r) · Y_lm(Ω) is theoretically derived. The corresponding method can be easily applied to any point group, including the molecular continuous groups C_∞v and D_∞h. On the basis of this (theoretical) lower bound, the efficiency of the permutational approach in generating sets of independent integrals is discussed. It is proved that lobe orbitals are always more efficient than the familiar Cartesian Gaussians, in the sense that GLO s provide the shortest integral lists. Moreover, it appears that the new axial GLO s often lead to a number of integrals, which is the theoretical lower bound previously defined. With AGLO s, the numbers of two-electron integrals to be computed, stored, and processed are divided by factors 2.9 (NH₃), 4.2 (C₅H₅), and 3.6 (C₆H₆) with reference to the corresponding CGTO s calculations. Remembering that in the permutational approach, atomic integrals are directly computed without any four-indice transformation, it appears that its utilization in connection with AGLO s provides one of the most powerful tools for treating symmetrical species.     

Molecular orbital studies of dinitrogen tetroxide and related molecules

Hartree-Fock LCAO MO calculations for N₂O₄ have been performed in a basis of symmetry orbitals formed from a minimal Slater basis set. Effects of rounding and truncation errors were minimized by the use of the symmetry basis, which also allowed the order or tilling of molecular orbitals to be specified independently of orbital energies. Convergence difficulties were overcome by combined use of the conjugate gradients method and Roothaan's iterative procedure; the method of steepest descents was less effective than either of these. Multicentre ‘non-NDDO’ two-electron integrals were evaluated by the gaussian expansion technique. The wavefunction obtained for the lowest state is NN antibonding, largely as a result of the filling of the 6b_1u antibonding sigma orbital in preference to the 6a_g bonding sigma orbital. There is only a small amount of NN pi-bonding. A bond energy analysis shows that the lowest state is markedly stabilized by NNO three-centre interactions.     

Calculations using a set of evenly spaced S-type gaussian functions

A basis set of evenly spaced S-type Gaussian functions with common exponents is examined. Formulas for common one- and two-electron integrals are derived. Because of thesymmetry of this basis set, a very compact two-electron integral list is produced. The number of two-electron integrals that must be stored is approximately eight times the number of basis functions. Use of this basis set in an SCF calculation is examined. Numerical results show that this approach works well for molecules containing only small atoms such as hydrogen, helium, or lithium, but that the method has problems with the core orbitals of heavier atoms. Procedures for augementing this basis set in calculations involving heavier atoms are examined.     

Two-electron repulsion integrals over Gaussian s functions

We present an efficient scheme to evaluate the [ 0 ]^(m) integrals that arise in many ab initio quantum chemical two-electron integral algorithms. The total number of floating-point operations (FLOPS ) required by the scheme has been carefully minimized, both for cases where multipole expansions of the integrals are admissable and for cases where this is not so. The algorithm is based on the use of a modified Chebyshev interpolation formula to compute the function exp(?T) and the integral F_m(T) = ∫₀¹u^2mexp(?Tu²) du very cheaply.     

Spectrum of states in icosahedral structures of the electronic configuration gn (N = 1–7). 1. Invariant expansion for integrals and energies

The energy spectrum of the states that appear in structures of icosahedral (I,I_h symmetry with open electronic shells g^N (dim g = 4; N = 1–7) is reported. The energies are obtained in terms of integral invariants (reduced matrix elements of electron-electron interaction) H^k (g, g). The latter are analogs of the Slater-Condon parameters F^k(l,l) for atoms with the l^N electronic configuration. A similar representation is proposed for the integrals mm’≨’) of electron-electron interaction on the 4-fold degenerate g orbitals in the “standard” representation. The relation between the terms of the g^N(I,I_h) configuration and the parent states of the orthogonal group O⁺(4) is discussed. Translated fromZhurnal Strukturnoi Khimii, Vol. 38, No. 1, pp. 3–13, January–February, 1997.     

Application of modified Neumann expansion for analytical evaluation of two-center,two-electron integrals over slater-type orbitals

A modified form of the Neumann expansion in terms of products of orthogonal polynomials for the inverse interelectronic distance r¹₁₂ is proposed. This expansion has been applied in order to derive a unified analytical formula for two-center and two-electron integrals over Slater-type orbitals. The results are equivalent to those given recently by Yasui and Saika, but the expansion itself can be used for building up a realistic algorithm for evaluation of three- and four-electron integrals determined by using correlated variational wave functions.     

Multiconfigurational SCF approach for high-symmetry molecules and its applications to fullerene trianion

A symmetry-adapted multiconfiguration self-consistent field (MC SCF) approach aimed at calculations of high-symmetry molecules is proposed. The self-consistency procedure applicable to the molecular terms of any symmetry and multiplicity is developed. It holds the symmetry transformation properties of varied molecular orbitals, thus taking advantage of the relationships within the set of two-electron integrals through molecular invariants. For orbital optimization, a unified coupling operator is constructed on the basis of the pseudosecular method providing for efficient convergence to energy minimum. Based on the group-theory technique, computer codes have been developed for straightforward determination of the invariant expansions for two-electron integrals and configuration interaction (CI) matrix elements. Calculated in this way, the expansion coefficients are presented for the three-electron states that originate from joint t_1u and t_1g shells of an icosahedral fullerene C₆₀, the case important for the calculations of anion C₆₀³⁻ representing the charge state of the fullerene molecule in the superconducting ionic solids K₃C₆₀ or Rb₃C₆₀. The results of MC SCF calculations for lowest quasi-π-electronic states of C₆₀³⁻ are discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 293–304, 1998     

Optimization of virtual orbitals in the framework of a multiconfiguration spin-coupled wave function

A new method is introduced for the optimization of nonorthogonal virtual orbitals for use in general multiconfiguration spin-coupled wave functions. The use of a number of highly effective approximations greatly reduces the computational effort involved, the most important being the use of a second-order perturbation expression for the energy and an approximate expression for the elements of the Hessian. As a result, the overall scheme scales very favourably with respect to the numbers of active electrons and of basis functions, making it suitable for the accurate study of large systems. Benchmark calculations are presented for the dissociation of LiH(X¹Σ⁺) and Li₂(X¹Σ⁺ _g ) using a highly compact four-configuration wave function. Standard spin-coupled valence bond expansions in the same virtual space are required to be significantly larger before equivalent results are obtained. The results are shown to compare very favourably with full valence complete active space self-consistent field calculations using an identical basis, and binding energies are within 4% of the values obtained from full configuration interaction calculations in the same basis set. Received: 10 June 1997 / Accepted: 7 October 1997     

A CNDO-MO calculation of VCl4

The electronic structure of the tetrahedral molecule VCL₄ is investigated within the CNDO-MO approximations. The metal and ligand valence orbitals, 3d, 4s, 4p; and 3s, 3p; respectively, have been systematically varied in an attempt to minimize the total energy; “optimum” V 4s(χ₄ = 1.10) and 4p(d ³ p ²) orbitals have been established, but V 3d(d ⁿ ) and Cl(-δ) valence orbitals are only seen to favor lower energy for expanded orbitals. Since determining the one-electron molecular orbital level which is occupied by the vanadium lone electron is a major aspect of this investigation, all calculations have been performed in triplicate: calculations assuming the unpaired electron occupies the 3a ₁, 2 _e and 4t ₂ molecular orbital (ground state electronic configurations² A ₁,² E, and² T ₂, respectively). The Hartree-Fock equations have been solved by Roothaan's SCF method for open shells, but off-diagonal multipliers between filled and partly filled molecular orbitals of the same symmetry have been neglected. As a qualitative estimate of the error introduced by this simplification, the pertinent overlap integrals between the eigenfunctions from calculations for the three possible configurations,² A ₁,² E, and² T ₂, are investigated as functions of the component 3d(d ⁿ ) and Cl(-δ) valence orbitals. The overlap integrals from the relevant² A ₁ and² T ₂ calculations are reasonably small, but the neglect of off-diagonal multipliers in calculations on the² E state is found to be a poor approximation. An ordering of the non-filled molecular orbitals in VCl₄ of 4t ₂ < 3a ₁ < 2e < 5t ₂ seems most consistent with the numerous calculations. This suggested ground state electronic configuration of² T ₂ introduces new aspects to the consideration of a (dynamic) Jahn-Teller effect in VCl₄. Experimental data pertinent to the electronic structure of VCl₄ has been briefly summarized, but unfortunately it is inadequate to confirm or deny the present calculations.     

Optical Study of Radicals (OH, O, H, N) in a Needle-plate Negative Pulsed Streamer Corona Discharge

Optical emission spectroscopy has been applied to study the OH radicals and O, H, and N active atoms produced by a high-voltage negative pulsed streamer corona discharge of N₂ and H₂O mixture gas in a needle-plate reactor at one atmosphere. The relative vibrational populations and the vibrational temperature of N₂(C, v′) were determined. The effects of pulsed peak voltage, pulsed repetition rate, and the addition of O₂ on the relative populations of OH(A²Σ) radicals, O(3p⁵P), H_α (3P), and N(3p⁴P) active atoms were investigated. It was found that the relative populations of those radicals increase with increasing pulsed peak voltage and pulsed repetition rate. The relative population of OH(A²Σ) radicals decreases with increasing O₂ flow rate, while the relative populations of O (3p⁵P), H_α (3P), and N (3p⁴P) active atoms exhibit a maximum over the studied range of the O₂ flow rate. The involved physicochemical processes have also been discussed.     

Elimination of the diagonalization bottleneck in parallel Direct-SCF methods

Summary It is shown that the matrix diagonalization bottleneck associated with thesequential O(N _BFN ³ ) diagonalization of the fock matrix within each iteration of the Direct-SCF procedure may be eliminated, and replaced instead with a combination ofparallel O(N _BFN ^<4 ) andsequential O(N _Sub ³ ) steps. For large basis sets, the relation N_Sub N_BFN between the dimension of the expansion subspace and the number of basis functions leads to a method of wave-function optimization in which the sequential bottleneck is eliminated. As a side benefit, the second-order iterative procedure on which this method is based displays superior convergence properties, and provides greater insight into the behavior of the energy with respect to orbital variations, than the traditional first-order, fixed-point, iterative approaches. The implementation of this method may be incorporated into essentially any existing Direct-SCF program with only minimal, and localized, changes.     

Scaling the Coulomb interaction in calculations of electron spectra of transition metal complexes

A simple technique of scaling two-electron integrals in ab initio calculations of the electronically excited states of transition metal complexes is proposed. This technique uses the fact that one-center two-electron integrals depend linearly on the scaling factor when Slater type functions are subjected to scaling transformation. This leads to a linear dependence of the d—d transition energy on the “scale” of Coulomb interaction, which allows one to affect the calculation result by varying the Slater exponential. To test the technique, ab initio configuration interaction and full active space calculations of the low excited states of the CrF ₆ ^3- , MnF ₆ ^2- , and VF ₆ ^3- complexes are performed. For transition elements, a basis of Slater type effective functions chosen from the optical spectra of the atoms and ions of transition elements is used. It is shown that in the STO-6G basis with effective exponentials, experimental transitions are reproduced with an accuracy of about 2000 cm^-1 even with the use of small active space determined by the orbitals localized on the central atom of the complex.     

The inclusion of a correlation function in the calculation of the ZFS of benzene

We calculated the one-center atomic integral of the spin—spin interaction with respect to Slater orbitals where we have included a two-electron correlation function that we used previously to calculate the hyperfine splitting of the 2³P state of He. The ZFS of the benzene molecule was calculated from this value of the one-center integral and by using the Mulliken approximation combined with a previously introduced approximation for extimating the other integrals. The calculation was performed for two different non-hexagonal configurations I and II, the results are D₁ = 0.132 cm^?1 and D_II = 0.143 cm^?1. The experimental value is D = 0.158 cm^?1.     

A comparative study of the aromaticity in some typical conjugated six-membered rings by the method of localized molecular orbital

The localized molecular orbitals and energy levels for four typical conjugated six-membered ring systems C₆H₆, C₃N₃H₃, B₂N₈H₄, and (B₂O₄)^3- as well as a non-aromatic reference molecule P_a-N₃Cl₆ have been calculated by using Edmiston- Ruedenberg energy localization technique under the CNDO / 2 approximation in order to investigate the nature of aromaticity or quasi-aromaticity of the six-membered ring systems studied. The contour maps for x-type localized MO's (LMO) have been plotted to illustrate the bonding characteristics of the five ring systems studied. These LMO calculations show that for all the conjugated six-membered ring systems considered there exists local delocalization of x-bonds or three-centered and occasionally four-centered two-electron x-bonds in our terminology, and the cooperative effect among these x-bonds leading to the formation of a closed continuous x-conjugation system around the ring, which is necessary for the creation of aromaticity in the systems studied. We have been able to discuss the properties of these three-centered x-bonds in terms of the constituent atoms and electrons and the relevant orbitals involved.     

Efficient use of Jacobi rotations for orbital optimization and localization

Summary Quantum chemical orbital optimizations can be accomplished by Newton-type iterations, whereall orbitals are improved ateach step; or by a succession of Jacobi rotations, where onlytwo orbitals are improved at any one step. In both schemes, the iterative updating of the four-index two-electron integrals requires a large computational effort. We show that the four-index transformation due to a Jacobi rotation can be simplified to such a degree that the successive execution of the four-index transformations ofN(N–1)/2 different Jacobi rotations requires no greater computational effort than that required by theone full orthogonal transformation which is the product of allN(N–1)/2 Jacobi rotations. The four-index updating has therefore no bearing on the relative merit of the Newton approach versus the Jacobi approach. The Jacobi approach has, however, an advantage if the optimization of each Jacobi rotation angle is simple and if the effectiveness of the individual Jacobi rotations can be assessed without the execution of four-index transformations. For, in that case, all ineffectual rotations are easily deleted from the iterative sequence. Whether convergence can be guaranteed for one or the other approach is also relevant. Our conclusions are illustrated by application to the problem of intrinsic orbital localization where the succession of Jacobi rotations is the more effective strategy.Dedicated to Professor Per Olov LöwdinAmes Laboratory is operated for the U.S. Department of Energy by Iowa State University under contract No. W-7405-Eng-82. This work was supported by the Division of Chemical Sciences, Office of Basic Energy Sciences     

Non-integer Slater orbital calculations

In order to calculate the one- and two-electron, two-center integrals over non-integer n Slater type orbitals, use is made of elliptical coordinates for the monoelectronic, hybrid, and Coulomb integrals. For the exchange integrals, the atomic orbitals are translated to a common center. The final integration is performed by Gaussian quadrature.As an example, an SCF ab initio calculation is performed for the LiH molecule, both with integer and non-integer principal quantum number.     

Influence of the Excited States of Atomic Nitrogen N(<Superscript>2</Superscript>D), N(<Superscript>2</Superscript>P) and N(R) on the Transport Properties of Nitrogen. Part II: Nitrogen Plasma Properties

In this paper, the calculated values of the viscosity and thermal conductivity of nitrogen plasma are presented taking into account five (e, N, N⁺, N₂ and N₂⁺) or eight (e, N(⁴S), N(²P), N(²D), N(R), N⁺, N₂ and N₂⁺) species. The calculations are based on the supposition that the temperature dependent probability of occupation of the states is given by the Boltzmann factor. The domain for which the calculations are performed, is for p = 1 and 10 atm in the temperature range from 5,000 K to 15,000 K. Classical collision integrals are used in calculating the transport coefficients and we have introduced new averaged collision integrals where the weight associated at each interacting species pair is the probable collision frequency. The influence of the collision integral values and energy transfer between two different species is studied. These results are compared which those of published theoretical studies.