Second and third derivatives of variational energy expressions: Application to multiconfigurational self‐consistent field wave functions

General analytical expressions are given for the second and third derivatives of constrained variational energy expressions. It is pointed out that variational energy expressions and odd‐order derivatives have a distinct advantage over nonvariational (e.g., perturbative) energy expressions and even‐order derivatives. In particular, the first‐order wave function suffices to determine the derivatives of the variational energy up to third order. The coupled‐perturbed multiconfigurational SCF (MC‐SCF) equations, obtained from the general results, are equivalent, with minor corrections, to the ones very recently presented by Osamura, Yamaguchi, and Schaefer. Explicit expressions are given for the second and third derivatives of the MC‐SCF energy. Computational implementation is briefly discussed.

[1]  P. Pulay.  Improved SCF convergence acceleration , 1982 .

[2]  B. Roos,et al.  A new method for large-scale Cl calculations , 1972 .

[3]  Ian M. Mills,et al.  Force Constants and Dipole-Moment Derivatives of Molecules from Perturbed Hartree-Fock Calculations. I , 1968 .

[4]  M. Dupuis.  Energy derivatives for configuration interaction wave functions , 1981 .

[5]  Peter Pulay,et al.  Systematic AB Initio Gradient Calculation of Molecular Geometries, Force Constants, and Dipole Moment Derivatives , 1979 .

[6]  Henry F. Schaefer,et al.  Gradient techniques for open‐shell restricted Hartree–Fock and multiconfiguration self‐consistent‐field methods , 1979 .

[7]  Hans-Joachim Werner,et al.  A quadratically convergent multiconfiguration–self‐consistent field method with simultaneous optimization of orbitals and CI coefficients , 1980 .

[8]  Byron H. Lengsfield,et al.  General second order MCSCF theory: A density matrix directed algorithm , 1980 .

[9]  Peter Pulay,et al.  Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules , 1969 .

[10]  Michel Dupuis,et al.  Molecular symmetry. II. Gradient of electronic energy with respect to nuclear coordinates , 1978 .

[11]  P. Pulay.  Convergence acceleration of iterative sequences. the case of scf iteration , 1980 .

[12]  E. Davidson.  The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices , 1975 .

[13]  J. Pople,et al.  Derivative studies in configuration–interaction theory , 1980 .

[14]  H. Schaefer,et al.  Generalization of analytic configuration interaction (CI) gradient techniques for potential energy hypersurfaces, including a solution to the coupled perturbed Hartree–Fock equations for multiconfiguration SCF molecular wave functions , 1982 .

[15]  Bernard R. Brooks,et al.  Analytic gradients from correlated wave functions via the two‐particle density matrix and the unitary group approach , 1980 .

[16]  P. Pulay,et al.  Cubic force constants and equilibrium geometry of methane from Hartree–Fock and correlated wavefunctions , 1978 .

[17]  Michel Dupuis,et al.  Evaluation of molecular integrals over Gaussian basis functions , 1976 .

[18]  G. Das.  A quadratically convergent MCSCF scheme using Fock operators , 1981 .

[19]  P.-O. Loewdin,et al.  Studies in Perturbation Theory. IX. Connection Between Various Approaches in the Recent Development—Evaluation of Upper Bounds to Energy Eigenvalues in Schrödinger's Perturbation Theory , 1965 .

[20]  A. Komornicki,et al.  An efficient ab initio method for computing infrared and Raman intensities: Application to ethylene , 1979 .

[21]  Klaus Ruedenberg,et al.  MCSCF optimization through combined use of natural orbitals and the brillouin–levy–berthier theorem , 1979 .

[22]  R. Moccia.  Variable bases in SCF MO calculations , 1970 .

[23]  Keiji Morokuma,et al.  Energy gradient in a multi-configurational SCF formalism and its application to geometry optimization of trimethylene diradicals , 1979 .

[24]  A. J. Sadlej,et al.  Coupled multiconfiguration self-consistent field (MC SCF) perturbation theory , 1975 .