Functional form of the wave function

In both variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC) (for a review about DMC see Ref. (26)) the trial-function completely determines the quality of the approximation for the physical observables. Because of this, it is extremely important to choose carefully a flexible wave-function that contains as much knowledge as possible of the physics of the system being studied.
In the first part of this thesis we proposed an highly correlated wave function that is able to capture the major part of the correlation energy: The Antisymmetric Geminal Product supplemented by the Jastrow correlation (JAGP). This wave function is an extension of the Antisymmetric Geminal Product AGP, introduced in quantum chemistry by Coleman (32):

$$\Psi_{AGP} (r_1,...,r_N)= \hat A \Pi_{i=1}^{N/2} \Phi(r_{2i},r_{2i-1})$$ (1.15)

where $\hat A$ is the antisymmetrization operator. The AGP wave-function is determined by the geminal, which is usually expanded in a one-particle basis:

$$\Phi(r_{i},r_{j}) = \sum _{1 \leq l,m \leq r} \lambda_{lm} \phi_l(r_{i}) \phi_m(r_{j})$$ (1.16)

where $r$ is the size of the orbital basis set. The geminal is then determined by $r(r-1)/2$ coefficients $\lambda$ . For instance, for the simple hydrogen molecules, using only two orbitals as basic set, the AGP is:

$$\Psi_{H_2} = \lambda_{11} \phi^A_{1s}(r_{1}) \phi^A_{1s}(r_{2}) +... ...(r_{1}) \phi^B_{1s}(r_{2}) + \lambda_{21} \phi^B_{1s}(r_{1}) \phi^A_{1s}(r_{2})$$ (1.17)

where $\Psi_{H_2}$ contains bonding and anti-bonding orbitals and so it is able to reproduce the Heithler-London limit. Notice that AGP wave-function is similar to the Gutzwiller BCS wave function used on lattice system (33).
The full JAGP wave-function is defined by the product of different terms, namely one-body, two-body, three-body Jastrow $J_1,J_2,J_3$ and an antisymmetric part ( $\Psi=J \Psi_{AGP}$ ). The first term is used to satisfy the nuclear cusp conditions, while the second the electron-electron one. The third one is an explicit contribution to the dynamic electronic correlation, and the latter is able to treat the non-dynamic one arising from near degenerate orbitals through the geminal expansion. Therefore our wave function is highly correlated and it is expected to give accurate results on widely range of systems.