Pairing determinant

As it is well known, a simple Slater determinant provides the exact exchange electron interaction but neglects the electronic correlation, which is by definition the missing energy contribution. In the past different strategies were proposed to go beyond Hartee-Fock theory. In particular a sizable amount of the correlation energy is obtained by applying to a Slater determinant a so-called Jastrow term, that explicitly takes into account the pairwise interaction between electrons.
On the other hand, within the Quantum Chemistry community the Antisymmetric Geminal Product (AGP) is a well known ansatz to improve the HF theory, because it implicitly includes most of the double-excitations of an HF state.

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 $N$ electrons (the first $N/2$ coordinates are referred to the up spin electrons) the AGP wave function is a $\frac{N}{2} \times \frac{N}{2}$ pairing matrix determinant, which reads:

$$\Psi_{AGP}(\vec{r}_1,...,\vec{r}_N) = \det \left (\Phi_{AGP}(\vec{r}_i,\vec{r}_{j+N/2}) \right ).$$ (1.18)

Here the geminal function is expanded over an atomic basis:

$$\Phi_{AGP}(\vec{r}^\uparrow,\vec{r}^\downarrow) =\sum_{l,m,a,b}{\lambda^{l,m}_{a,b}\phi_{a,l} (\vec{r}^\uparrow)\phi_{b,m}(\vec{r}^\downarrow)} ,$$ (1.19)

where indices $l,m$ span different orbitals centred on atoms $a,b$ , and $i$ ,$j$ are coordinates of spin up and down electrons respectively.
Differently from the previous pairing function formulation (2), appropriate only for simple atoms, here also off-diagonal elements are included in the $\lambda$ matrix, which must be symmetric in order to define a spin singlet state. Moreover this formulation allows to easily fulfill other symmetries by imposing the appropriate relations among different $\lambda_{l,m}$ . For instance in homo-nuclear diatomic molecules, the invariance under reflection in the middle plane perpendicular to the molecular axis yields the following relation:

$$\lambda^{a,b}_{m,n}=(-1)^{p_m+p_n} \lambda^{b,a}_{m,n},$$ (1.20)

where $p_m$ is the parity under reflection of the $m-$ th orbital.

An important property of this formalism is the possibility to describe explicitly resonating bonds present in many structures, like benzene. A $\lambda^{a,b}_{m,n}$ 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 $\lambda^{a,b}_{m,n}$ coefficients: the relative weight of each bond is related to the amplitude of the corresponding $\lambda$ .

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 $A$ and $B$ , each containing a given number of atoms, the many-electron wave function $\Psi$ factorizes into the product of space-disjoint terms $\Psi = \Psi_A \bigotimes \Psi_B$ as long as the interaction between the electrons coupling the different regions $A$ and $B$ 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 $a$ if the following linear system is satisfied:

$$\sum^{(1s,2s)}_{j}{\lambda^{j,j^\prime}_{a,b} \hat \phi'_{a,j}(\t... ..._a \sum_{c,j}{\lambda^{j,j^\prime}_{c,b}\phi_{c,j} (\textbf{r}=\textbf{R}_a)} ,$$ (1.21)

for all $b$ and $j^\prime$ ; in the LHS the caret denotes the spherical average of the orbital gradient. If we impose that the orbitals satisfy the atomic cusp condition on their atom, this equation reduce to:

$$\sum_{c(\neq a),j}{\lambda^{j,j'}_{c,b}\phi_{c,j}(\textbf{R}_a)} = 0,$$ (1.22)

and because of the exponential orbital damping, if the nuclei are not close together each term in the previous equations is very small, of the order of $\exp (-\vert\textbf{R}_a-\textbf{R}_c\vert)$ . Therefore in the first part of this thesis, with the aim of making the optimization faster, we have chosen to use $1s$ and $2s$ orbitals satisfying the atomic cusp conditions and to disregard the sum (1.22). In this way, once the energy minimum is reached, also the molecular cusp conditions (1.21) are rather well satisfied. Later in the second part of the thesis we have adopted a different and more efficient strategy to the cusp problem as described in the following section.