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Integrals and derivatives for correlated Gaussian functions using matrix differential calculus
Author(s) -
Kinghorn Donald B.
Publication year - 1996
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/(sici)1097-461x(1996)57:2<141::aid-qua1>3.0.co;2-y
Subject(s) - differential calculus , kronecker delta , matrix (chemical analysis) , time scale calculus , mathematics , matrix function , pascal matrix , matrix calculus , gaussian , kronecker product , hamiltonian (control theory) , block matrix , notation , functional calculus , algebra over a field , multivariable calculus , pure mathematics , symmetric matrix , quantum mechanics , physics , eigenvalues and eigenvectors , chemistry , mathematical optimization , chromatography , control engineering , engineering , arithmetic
The matrix differential calculus is applied for the first time to a quantum chemical problem via new matrix derivations of integral formulas and gradients for Hamiltonian matrix elements in a basis of correlated Gaussian functions. Requisite mathematical background material on Kronecker products, Hadamard products, the vec and vech operators, linear structures, and matrix differential calculus is presented. New matrix forms for the kinetic and potential energy operators are presented. Integrals for overlap, kinetic energy, and potential energy matrix elements are derived in matrix form using matrix calculus. The gradient of the energy functional with respect to the correlated Gaussian exponent matrices is derived. Burdensome summation notation is entirely replaced with a compact matrix notation that is both theoretically and computationally insightful. © 1996 John Wiley & Sons, Inc.