Numerical instabilities in the computation of pseudopotential matrix elements

Steep high angular momentum Gaussian basis functions in the vicinity of a nucleus whose inner electrons are replaced by an effective core potential may lead to numerical instabilities when calculating matrix elements of the core potential. Numerical roundoff errors may be amplified to an extent that spoils any result obtained in such a calculation. Effective core potential matrix elements for a model problem are computed with high numerical accuracy using the standard algorithm used in quantum chemical codes and compared to results of the MOLPRO program. Thus, it is demonstrated how the relative and absolute errors depend an basis function angular momenta, basis function exponents and the distance between the off-center basis function and the center carrying the effective core potential. Then, the problem is analyzed and closed expressions are derived for the expected numerical error in the limit of large basis function exponents. It is briefly discussed how other algorithms would behave in the critical case, and they are found to have problems as well. The numerical stability could be increased a little bit if the type 1 matrix elements were computed without making use of a partial wave expansion.     

On the rapid evaluation of cofactors in the calculation of nonorthogonal matrix elements

The calculation of matrix elements involving nonorthogonal orbitals is speeded up by recognizing the orthogonalities between orbitals, leading to generalized Slater rules. The block structure present in the overlap matrix makes an efficient evaluation of its cofactors possible. These cofactors are calculated per subblock, each with its own parity sign. An adjustment parity sign has to be evaluated, which is added to the combined local signs, to give the correct total sign for the matrix element. An algorithm for the evaluation of this adjustment sign has been developed, making an easy and correct evaluation possible. The current scheme is shown to be very efficient, but possibilities for further improvement remain. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 77–83, 1998     

On the computation of matrix elements between numerical wave functions: The canonical functions method

The problem of the computation of the matrix elements is considered when Ψ_v(r) and Ψ_v(r) are eigenfunctions related to a diatomic potential of the RKR type (defined by the coordinates of its turning points P_i with polynomial interpolations). The eigenfunction Ψ(r) is computed by the canonical functions method making use of the abscissas r_i of P_i uniquely. This limited number of points allows the storage of ψ_v(r_i) for all the required levels v, and reduces greatly the computational effort when v, ν′, and k are varying. The present method maintains all the advantages of a highly accurate numerical method (even for levels near the dissociation), and reduces greatly the computing time. Furthermore, it is shown that it may be extended to analytical potentials like Morse and Lennard-Jones functions, to vibrational-rotational eigenfunctions and to matrix elements between eigenfunctions related to two different potentials. Numerical applications are presented and discussed.     

On the transferability of Fock matrix elements

The transferability of Fock matrix elements in the linear combination of atomic orbitals molecular orbital scheme is analysed using localized orbitals. It is shown that this transferability is dependent on the transferability of these localized orbitals and the neglect of long-range contributions from partially cancelling Coulomb nuclear attraction and electron repulsion terms. A theoretical basis is thus provided for the simulated ab initio molecular orbital and related methods. Various corrections previously introduced in an ad hoc manner are shown to be justified. Transferability in both the closed shell and open shell schemes is analysed.     

On the calculation of vibronic coupling matrix elements

The non-diagonal matrix elements in the adiabatic Born-Oppenheimer approximation are considered. The effect of the Q-dependence of the electronic energy denominator is calculated explicitly for an arbitrary initial and final state. It is shown that the inclusion of this effect does not change the relative values of the coupling matrix elements for different initial vibronic states.     

On the summations involving Wigner rotation matrix elements

Two new methods to evaluate the sums over magnetic quantum numbers, together with Wigner rotation matrix elements, are formulated. The first is the coupling method which makes use of the coupling of Wigner rotation matrix elements. This method gives rise to a closed form for any kind of summation that involves a product of two Wigner rotation matrix elements. The second method is the equivalent operator method, for which a closed form is also obtained and easily implemented on the computer. A few examples are presented, and possible extensions are indicated. The formulae obtained are useful for the study of the angular distribution of the photofragments of diatomic and symmetric-top molecules caused by electric-dipole, electric-quadrupole and two-photon radiative transitions. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the closed form of Wigner rotation matrix elements

The closed forms of some rotation matrix elementsd _{m m} ^j (/2) are presented. The closed forms of summation involved two binomials and some special hypergeometric functions are also obtained. The MAPLE V program which calculates d _{m m} ^j (), dm,. (/2) and the help file are given in appendix.Supported in part by the Chiu Feng-Chia research Fund. Thanks are due to the Board of Trustees Chairman Dr. Ying-Ming Liao as well as the Trustee Dr. Charles Chiu-Hsiong Huang and President Dr. Ted C. Yang of Feng-Chia University, Taichung, Taiwan, ROC.     

Linear scaling computation of the Fock matrix. VII. Parallel computation of the Coulomb matrix

We present parallelization of a quantum-chemical tree-code for linear scaling computation of the Coulomb matrix. Equal time partition is used to load balance computation of the Coulomb matrix. Equal time partition is a measurement based algorithm for domain decomposition that exploits small variation of the density between self-consistent-field cycles to achieve load balance. Efficiency of the equal time partition is illustrated by several tests involving both finite and periodic systems. It is found that equal time partition is able to deliver 91%-98% efficiency with 128 processors in the most time consuming part of the Coulomb matrix calculation. The current parallel quantum chemical tree code is able to deliver 63%-81% overall efficiency on 128 processors with fine grained parallelism (less than two heavy atoms per processor).     

A method for the rapid calculation of matrix elements with highly oscillatory JWKB radial wavefunctions

A method for the rapid computation of matrix elements with JWKB radial wavefunctions is discussed. The method consists of dividing the range of integration into segments determined by the nodes of the semiclassical wavefunction. The desired matrix elements are calculated by summing the contributions from each segment which are evaluated by integrating between nodes with a Gauss-Mehler quadrature formula. The results are compared with exact quantum mechanical calculations and were found to agree within 1–2%. The calculations with the present method were generally five to ten times faster than the quantum mechanical calculations.     

On the evaluation of CI matrix elements for a canonically ordered basis

Using the unitary group approach it is shown that the amount of storage needed for the construction of symbolic CI matrix element lists for N-electron basis functions with large numbers of open shells and arbitrary multiplicities may substantially be reduced compared to methods currently available in the literature.     

On the evaluation of non-adiabatic coupling matrix elements for large scale CI wavefunctions

The implementation of a recently proposed technique for evaluating matrix elements of the form 〈Ψ_J(r;R)|∂Ψ_I(r;R)/t6R_α〉_r using analytic gradient techniques is described. The Ψ_K(r;R) are developed from state-averaged multiconfiguration self-consistent-field/configuration interaction (CI) wavefunctions. The CI wavefunctions are determined using the shape driven graphical unitary group approach. The method is shown to be considerably more efficient than presently existing approaches based on divided differences. As an illustration of the potentialities of this approach non-adiabatic coupling matrix elements are determined for the collinear charge transfer reaction: Mg(¹S) + FH(¹Σ⁺)→MgF(²Σ⁺)+H(²S).     

On the calculation of expectation values and transition matrix elements by coupled cluster method

Summary Finite order expressions are derived for expectation values and transition matrix elements within the framework of the coupled cluster method.     

Symmetrization of operator matrix elements

The method of Dupuis and King for generating matrix elements of a totally symmetric one-electron operator in terms of symmetry-distinct integrals only is generalized to the case of nontotally symmetric operators. For operators constructed from two-electron integrals, explicit reduction of integral processing to permutationally inequivalent symmetry-distinct integrals only is described, while for one-electron operators further reductions are derived using double coset decompositions. Finally, some computational consequences of this approach are briefly discussed.     

Explicit formulas for the Hamiltonian matrix elements

The explicit formulas for the evaluation of the Hamiltonian matrix elements are presented. The calculation of the integral coefficients is independent of both the nature of the orbitals and th spin coupling schemes. It is fully automatic and only dependent on the number of doubly and singly occupied orbitals. Further-more, the symmetric group representation matrices are not needed, and the N! problem can be avoided.     

On the computation of large-scale self-consistent-field iterations

The computation of the subspace spanned by the eigenvectors associated to the N lowest eigenvalues of a large symmetric matrix (or, equivalently, the projection matrix onto that subspace) is a difficult numerical linear algebra problem when the dimensions involved are very large. These problems appear when one employs the self-consistent-field fixed-point algorithm or its variations for electronic structure calculations, which requires repeated solutions of the problem for different data, in an iterative context. The naive use of consolidated packages as Arpack does not lead to practical solutions in large-scale cases. In this paper we combine and enhance well-known purification iterative schemes with a specialized use of Arpack (or any other eigen-package) to address these large-scale challenging problems.     

On the computation of molecular electronic affinities

Using a multireferent MBPT method (CIPSI) the electronic affinity (EA) of F, CN and HCC is computed. Results show how UMP2 gives unbalanced truncation of the MP series, while ROMP2 has the correct (balanced) behaviour. The good agreement with the experimental EA found for some compounds is accidental and associated to an error compensation. The good agreement with the experimental data found for the ROMP2 and CIPSI EAs is analysed.This paper was presented at the International Conference on The Impact of Supercomputers on Chemistry, held at the University of London, London, UK, 13–16 April 1987     

Linear scaling computation of the Fock matrix. III. Formation of the exchange matrix with permutational symmetry

 A direct comparison is made between two recently proposed methods for linear scaling computation of the Hartree–Fock exchange matrix to investigate the importance of exploiting two-electron integral permutational symmetry. Calculations on three-dimensional water clusters and graphitic sheets with different basis sets and levels of accuracy are presented to identify specific cases where permutational symmetry may or may not be useful. We conclude that a reduction in integrals via permutational symmetry does not necessarily translate into a reduction in computation times. For large insulating systems and weakly contracted basis sets the advantage of permutational symmetry is found to be negligible, while for noninsulating systems and highly contracted basis sets a fourfold speedup is approached. Received: 8 October 1999 / Accepted: 3 January 2000 / Published online: 21 June 2000