Recently a new trial function was proposed for atoms, that includes both the terms (2). In the first part of this thesis we extend this promising approach to a number of small molecular systems with known experimental properties, that are commonly used for testing new numerical techniques.
The major advantage of this approach is the inclusion of many CI expansion terms with the computational cost of a single determinant. For instance this has allowed us to perform the full structural optimization of benzene without a particularly heavy computational effort on a single processor machine.
For an unpolarized system containing electrons (the first coordinates are referred to the up spin electrons) the AGP wave function is a pairing matrix determinant, which reads:
An important property of this formalism is the possibility to describe explicitly resonating bonds present in many structures, like benzene. A different from zero represents a chemical bond formed by the linear combination of the m-th and n-th orbitals belonging to a-th and b-th nuclei. It turns out that resonating bonds can be well described through the geminal expansion by switching on the appropriate coefficients: the relative weight of each bond is related to the amplitude of the corresponding .
Also polarized systems can be treated within this framework, by using the spin generalized version of the AGP (GAGP), in which also the unpaired orbitals are expanded as well as the paired ones over the same atomic basis employed in the geminal (34).
Another important property of AGP wave-function is the size consistency: if we smoothly increase the distance between two regions and , each containing a given number of atoms, the many-electron wave function factorizes into the product of space-disjoint terms as long as the interaction between the electrons coupling the different regions and can be neglected. In this limit the total energy of the wave function approaches the sum of the energies corresponding to the two space-disjoint regions. This property, that is obviously valid for the exact many-electron ground state, is not always fulfilled by a generic variational wave function as for instance configuration-interaction (CI) wave-function. Notice that this property is valid when both the compound and the separated fragments have the minimum possible total spin. This is precisely the relevant case for hydrogen phase diagram studied in this thesis because we have not studied ferromagnetic or partially ferromagnetic phases, that are not believed to be present in the reasonable pressure-temperature range of hydrogen (for a discussion about ferromagnetism in high-pressure hydrogen see Ref. (35)).
Now we want to highlight how it is possible to implement nuclear cusp condition (see Appendix C) for molecular systems with a pairing wave-function. A straightforward calculation shows that the AGP wave function fulfills the cusp conditions around the nucleus if the following linear system is satisfied: