Variational Interaction Terms

The general equations for the variational parameters in a
parameterized density matrix, from Eq. 3.19, are

and the norm matrix

(232) |

(233) |

Repeating Eq. 3.56, the parameterized variational density
matrix is an anti-symmetrized product of one-particle density matrices,

(234) | |||

(235) |

where the amplitude and the widths and means are the variational parameters. We also dropped prefactors which are the same for the norm matrix and thus cancel out. The permutation sum is over all permutations of identical particles (e.g. same spin electrons) and is the permutation signature. The initial conditions are , , and .

For this ansatz the generator of the norm matrix,

For a periodic system the above equation also is summed over all periodic simulation cell vectors, , with . Using this the components of the norm matrix are then:

(236) | |||

(237) | |||

(238) | |||

(239) | |||

(240) | |||

(241) |

The Hamiltonian for a periodic system of electrons and ions is given by,

(242) |

(243) |

where is the periodic cell volume and an arbitrary constant. The Madelung term in is the interaction energy of an electron with it's periodic images and neutralizing background (e.g. for a simple cubic simulation cell, the usual case). To do the integrals, we represent the Gaussians by their Fourier series

(244) |

(245) |

(246) | |||

where and . The interaction integral

is symmetric in when the periodic cell has inversion symmetry. Continuing, the left hand side of Eq. A.1 is

(248) | |||

(249) | |||

(250) |

with

(251) | |||

(252) | |||

(253) | |||

(254) |

where we have used the fact that terms in and give the same contribution under the permutation sum and so combined them. The derivatives of the interaction integral are,

where and denote the derivatives of with the first and second argument. Comparing equation A.18 and Eq. A.15 the interaction integral may be written as

(255) |

(256) |

For an isolated system ( ) and these would simplify to,

(257) | |||

(258) | |||

(259) |

At , the initial derivatives for the variational parameters reduce to

(260) | |||

(261) | |||

(262) |

For large numbers of electrons, the computational requirements to treat all exchange terms increase drastically. Here the approximation discussed in section 3.7 is used where the kinetic pair exchange corrections given there are added to the identity permutation term derived here.