TY - JOUR

T1 - Newton-Raphson approaches and generalizations in multiconfigurational self-consistent field calculations

AU - Yeager, Danny L.

AU - Lynch, Diane

AU - Nichols, Jeffrey

AU - Jørgensen, Poul

AU - Olsen, Jeppe

PY - 1982

Y1 - 1982

N2 - We develop and review multiconfiguration Hartree-Fock procedures based on unitary exponential operators. When the energy functional is expanded through second order in the orbital optimization operator, κ̂, and the state optimization operator, Ŝ, and the variational principle applied, the resulting set of coupled, linear inhomogeneous equations are known as the Newton-Raphson equations. These equations demonstrate quadratic convergence. Particularly when far from convergence, small eigenvalues of the Hessian may give extremely large step length amplitudes. We discuss two procedures, mode damping and mode controlling, which are used to reduce large step length amplitudes. With both procedures mode reversal is also used to assure that the proper number of negative eigenvalues is present for each iteration. New calculational results are given for the first excited 1Σg+ state of C2 and the 21A1 state of CH2. For an optimized MCSCF state the generalized Brillouin's theorem is satisfied. There may be other criteria which we want the MCSCF state to fullfil, e.g., the nth excited state of a certain symmetry have n negative eigenvalues of the Hessian. We discuss several such criteria, including that the MCTDHF not be unstable and that the MCTDHF should have n negative excitation energies for the nth excited state of a certain symmetry. Finally, generalizations of Newton-Raphson and the multiplicity independent Newton-Raphson (MINR) approaches are discussed. We show how fixed Hessian-type approaches will demonstrate quadratic, cubic, quartic, ... convergency. Calculations are presented for the E3Σu- state of O2. MINR approaches may be useful when small eigenvalues of the Hessian are present. Since the MINR formula involves the product of third-derivative matrices with vectors, explicit evaluation of the third-derivative matrices is avoided.

AB - We develop and review multiconfiguration Hartree-Fock procedures based on unitary exponential operators. When the energy functional is expanded through second order in the orbital optimization operator, κ̂, and the state optimization operator, Ŝ, and the variational principle applied, the resulting set of coupled, linear inhomogeneous equations are known as the Newton-Raphson equations. These equations demonstrate quadratic convergence. Particularly when far from convergence, small eigenvalues of the Hessian may give extremely large step length amplitudes. We discuss two procedures, mode damping and mode controlling, which are used to reduce large step length amplitudes. With both procedures mode reversal is also used to assure that the proper number of negative eigenvalues is present for each iteration. New calculational results are given for the first excited 1Σg+ state of C2 and the 21A1 state of CH2. For an optimized MCSCF state the generalized Brillouin's theorem is satisfied. There may be other criteria which we want the MCSCF state to fullfil, e.g., the nth excited state of a certain symmetry have n negative eigenvalues of the Hessian. We discuss several such criteria, including that the MCTDHF not be unstable and that the MCTDHF should have n negative excitation energies for the nth excited state of a certain symmetry. Finally, generalizations of Newton-Raphson and the multiplicity independent Newton-Raphson (MINR) approaches are discussed. We show how fixed Hessian-type approaches will demonstrate quadratic, cubic, quartic, ... convergency. Calculations are presented for the E3Σu- state of O2. MINR approaches may be useful when small eigenvalues of the Hessian are present. Since the MINR formula involves the product of third-derivative matrices with vectors, explicit evaluation of the third-derivative matrices is avoided.

UR - http://www.scopus.com/inward/record.url?scp=29144459651&partnerID=8YFLogxK

U2 - 10.1021/j100209a006

DO - 10.1021/j100209a006

M3 - Article

AN - SCOPUS:29144459651

SN - 0022-3654

VL - 86

SP - 2140

EP - 2153

JO - Journal of Physical Chemistry

JF - Journal of Physical Chemistry

IS - 12

ER -