^{1}

^{*}

^{2}

^{2}

The Hartree-Fock equation is non-linear and has, in principle, multiple solutions. The *ω ^{th}* HF extreme and its associated virtual spin-orbitals furnish an orthogonal base B

^{ω}of the full configuration interaction space. Although all B

^{ω}bases generate the same CI space, the corresponding configurations of each B

^{ω}base have distinct quantum-mechanical information contents. In previous works, we have introduced a multi-reference configuration interaction method, based on the multiple extremes of the Hartree-Fock problem. This method was applied to calculate the permanent electrical dipole and quadrupole moments of some small molecules using minimal and double, triple and polarized double-zeta bases. In all cases were possible, using a reduced number of configurations, to obtain dipole and quadrupole moments in close agreement with the experimental values and energies without compromising the energy of the state function. These results show the positive effect of the use of the multi-reference Hartree-Fock bases that allowed a better extraction of quantum mechanical information from the several B

^{ω}bases. But to extend these ideas for larger systems and atomic bases, it is necessary to develop criteria to build the multireference Hartree-Fock bases. In this project, we are beginning a study of the non-uniform distribution of quantum-mechanical information content of the B

^{ω}bases, searching identify the factors that allowed obtain the good results cited above

The repulsive coulomb term in the electronic Hamiltonian of atoms, molecules and solids introduces a hard difficult to determine stationary electronic states. It is possible only to obtain approximate solutions of the Schrö- dinger’s equation and there are several procedures for this goal. Two important and general approaches to achieve this purpose are the variational and perturbative methods. In this essay, we are primarily concerned with the approximate variational methods, in particular, with the configuration interaction (CI) methods.

An initial variational approach to study electronic structure of atoms and molecules is the Hartree-Fock (HF) approximation. This approximation can be used as starting-point for other variational and perturbative methods. Although the HF approximation provides the most part of the electronic ground state energy of the system, the rest of this energy is very important to understand several phenomena involving these systems. Rigorously, in practice it is impossible to determine the HF limit as well the exact electronic ground state energy because the atomic bases are finite. Thus, the quantity of correlation energy that is possible to rescue is limited by the atomic base and other truncations or approximations employed. But it is possible for some systems to obtain very accu- rate values. Beside this, an error in the electronic energy means an error in the correspondent state vector with consequences on the calculated values of other dynamical variables, in particular on the non-variational dynam- ical variables.

In molecular electronic problems, there is a variety of observables that depend of the charge distribution and that are not variational dynamical variables as, for instance, the electrical multipolar moments. Non-variational dynamical variables do not exhibit the decrescent monotonical behavior of the energy with the augmentation of the variational class of functions. In contrast, they display an irregular behavior. This means that in the scope of the approximate variational methods better values for electronic energy do not implicate in better values of non- variational observables. This introduces difficulties in the determination of non-variational properties in the ambit of these approximate methods. Over the last twenty years, multi-reference configuration interaction (MRCI) methods have been developed and used in molecular quantum mechanics to study this question [

The traditional CI method uses a single reference, the HF ground state, and your excited configurations to ex- pand the state function. However, the HF equation is non-linear and has, in principle, multiple solutions [

In previous works [

For the restricted HF (RHF) problem of a molecule with

In a molecular HF problem, formulated in any class of symmetry of point and spin, it is possible, in principle, to obtain several HF solutions. And from these solutions to construct several HF references and your corre- spondent excited configuration state functions (CSFs)

sub-bases

system of generators of the full CI space. From the set

The motivation for using the MRHF bases lies on the following consideration. Let

or as,

where,

However,

Thus, in spite of the fact that the bases

The MRHFCI method was applied to calculate the permanent electrical dipole moment of some small molecules, LiH, BH, FH, CO and H_{2}O and the quadrupole moment of the FH [

In the current stage of development of the MRHFCI method, the results are still obtained through trials but this requires too much computational effort. To extend these calculations for larger systems and atomic bases it is necessary to develop criteria to build the MRHF bases. With this purpose, we begin this project with a systemat- ic study of the non-uniform distribution of QMIC of the

HF Solution (energy) | S (10 CSFs) | D (75 CSFs) | T (160 CSFs) | Q (172 CSFs) | SD (85 CSFs) | ST (170 CSFs) | SQ (182 CSFs) | DT (235 CSFs) |
---|---|---|---|---|---|---|---|---|

−7.98066895 | −7.83037244 | −7.49762404 | −5.17255295 | −1.82147305 | −7.88150734 | −7.84252804 | −7.83037244 | −7.49927038 |

−7.42164900 | −7.83621873 | −7.97148207 | −5.61879432 | −2.32622279 | −7.99818789 | −7.84838259 | −7.83621873 | −7.97322343 |

−7.24880471 | −7.76737105 | −7.99349350 | −5.79862878 | −2.44994704 | −7.99815341 | −7.78005950 | −7.76737105 | −7.99464208 |

−5.46835770 | −6.80462160 | −7.96556261 | −5.80028937 | −2.47022170 | −7.99625134 | −6.81681649 | −6.80462160 | −7.96619330 |

−2.45215236 | −5.82542732 | −7.93265323 | −7.72473656 | −7.38442761 | −7.94549594 | −7.72480279 | −7.38442761 | −7.98869674 |

−1.98791705 | −5.43622153 | −7.99049201 | −5.85354486 | −2.52779005 | −7.99916590 | −5.85451692 | −5.43622153 | −8.00352210 |

−1.87200145 | −5.57392280 | −7.87428572 | −7.71559905 | −7.43121356 | −7.87586037 | −7.71566426 | −7.43121356 | −7.99702073 |

−1.76533380 | −5.60479281 | −7.86014360 | −7.72955310 | −7.52152511 | −7.86014989 | −7.72955381 | −7.52152511 | −7.99444827 |

−1.50719389 | −5.06697657 | −7.33252181 | −7.81031691 | −7.97125621 | −7.33323310 | −7.81032990 | −7.97125621 | −7.82949725 |

0.14654857 | −4.21817022 | −7.44762052 | −7.58965583 | −7.82509874 | −7.46468615 | −7.59009739 | −7.82509874 | −7.89838385 |

1.01516265 | −3.03166839 | −7.40509900 | −7.80968434 | −7.95495322 | −7.42436175 | −7.82024827 | −7.95495322 | −7.83341686 |

3.02497886 | −1.06524777 | −5.45191381 | −6.77217340 | −7.91630275 | −5.47245848 | −6.78218398 | −7.91630275 | −6.81208380 |

S, D, T and Q indicate single, double, triple and quadruple excited CSFs; Li–H distance: 3.015 bohr; All the HF solutions are

(b)

In the last column R indicates HF reference; S, D, T and Q indicate single, double, triple and quadruple excited CSFs; Li-H distance: 3.015 bohr; All the HF solutions are

CSFs | HF Solutions | MRHFCI Energy | Dipole Moment |
---|---|---|---|

52 | A, B | −8.01014966 | 5.8321 |

37 | A, D | −8.00889691 | 5.8259 |

71 | A, L | −8.01016736 | 5.8199 |

418 | Full CI | −8.01152345 | 5.7503 |

Total energies (electronic + nuclear repulsion) of the HF solutions: A = −7.98066895 hartree; B = −7.42164900 hartree; D = −5.46835770 hartree; L = 3.09788624 hartree; All the HF solutions are

CSFs | HF Solutions | MRHFCI Energy |
---|---|---|

25 | A, B, C | −14.61597562 |

102 | Full CI | −14.61597567 |

Energies of the HF solutions: A = −14.57125146 hartree; B = −13.78788831 hartree; C = −3.47143646 hartree; All the HF solutions are ^{1}S and were obtained using the new double-zeta base [

As observed in Section 2, in spite of the fact that the bases

A second step in this project, now in progress, is to study the composition of the CSFs of the system of generators

L. A. C. Malbouisson thanks Professor M. A. Chaer Nascimento (Instituto de Química—Universidade Federal do Rio de Janeiro—Brazil) for his hospitality during the written of this work.