Integrals and derivatives for correlated Gaussian functions using matrix differential calculus
✍ Scribed by Donald B. Kinghorn
- Publisher
- John Wiley and Sons
- Year
- 1996
- Tongue
- English
- Weight
- 973 KB
- Volume
- 57
- Category
- Article
- ISSN
- 0020-7608
No coin nor oath required. For personal study only.
✦ Synopsis
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 theorctically and computationally insightful. (cj 1996 John Wiley & Sons, Inc.
I n 1 rod iic lion ost problems in quantum chemistry can be M formulated in matrix form and often involve variational principles and, hence, matrix derivatives (sometimes with respect to matrix variables). Without an effective matrix calculus, differentiation of matrix equations leads to torturous masses of summation signs and indices which are Thi5 work is '1 component of the author's thesis. Correspondence t o the author should be directed care o f I<. [I. f'o\hu+tci a t Washington State University Department o f Chmii5try