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Introduction

The state of many-electron system is described by a wave function $\Psi$, that is the solution of the Schrödinger equation:
\begin{displaymath}
({\cal H}-E)\psi=0
\end{displaymath} (6)

The approximate wave function $\Psi$ of the state labeled $\gamma L S$ is:

\begin{displaymath}
\Psi ( \gamma L S) = \sum_j c_j \Phi (\gamma _j L S ),
\end{displaymath} (7)

where $\gamma$ represents the dominant configuration, and any additional quantum numbers required for uniquely specifying the state being considered. The MCHF wave function $\Psi$ is expanded in terms of configuration state functions (CSF) $\{ \Phi \}$ having the same $LS$ symmetry but arising from different electronic configurations ($ \gamma _j $). The mchf procedure consists of optimizing to self-consistency both the sets of radial functions $\{ P_{n_j l_j} (r) \} $ and mixing coefficients $\{ c_j \}$. The CSF's are built from a basis of one-electron spin-orbital functions and determine the radial functions:
\begin{displaymath}
\phi _{nlm_lm_s} = \frac{1}{r} P_{nl} (r) Y_{lm_l}
(\theta, \varphi) \chi_{m_s}.
\end{displaymath} (8)

With the wave function expansion is associated an energy functional for one LS term and eigenvalue.

The traditional mchf program has been extended to accomplish a simultaneous optimization of energy expressions derived from several different terms or even several eigenvalues of the same term. Additionally, the energy energy functional is represented as a weighted average of energy functionals for expansions of wave functions for different LS terms or parity. This approach facilitates the Breit-Pauli calculations for complex atomic systems, while previously somewhat arbitrary methods have been applied (cross-wise optimization,  [*]).

mchf was modified for systematic, large-scale methods using dynamic memory allocation and sparse matrix methods. All orbitals in a wave function expansion are assumed to be orthonormal. Configuration states are restricted to at most eight (8) subshells in addition to the closed shells common to all configuration states. The maximum size is limited by the available memory and disk space. The wave function expansions are obtained from orbital sets of increasing size, allowing for the monitoring of convergence. The Davidson algorithm [#!dvdson!#] is applied for finding the needed eigenvalues and eigenvectors. In this version of the code, non-orthogonality is not supported. In the present atsp2K_MCHF package, it is not foreseen that optimization would be over different parities, only over different terms of the same parity, and we refer to this as "simultaneous optimization". Suppose ${\cal E}(T_i)$ represents and energy functional for term $T$ and eigenvalue $i$, assuming orbitals and also wave functions are normalized. Then optimization was performed on the functional

\begin{displaymath}{\cal E} = \sum_{T_i} w_{T_i} {\cal E}(T_i) /\sum_{T_i}w_{T_i}\end{displaymath}

where $w_{T_i}$ is the weight for ${T_i}$.


next up previous contents
Next: Program Structure Up: MCHF Previous: MCHF   Contents
2001-10-11