Variational transcorrelated method

We propose a new approach to the use of Jastrow ansatz in the calculation of electron correlations, based on a modification of the transcorrelated method of Boys and Handy [Proc. R. Soc. London, Ser. A 309, 209 (1969)]. In this new method, the original transcorrelated orbital equation is replaced with a general variational equation for the reference wave function, whereas the equation for the correlation factor remains the same. The method can be applied to a single determinant Jastrow ansatz as well as to a multideterminant one. For the single determinant ansatz, we obtain a Hartree-Fock type self-consistent equation for the optimization of orbitals, and for the multideterminant ansatz we have tested a CI type equation. We apply the new method in calculations of the C(2) molecule and compare the results with those of variational quantum Monte Carlo calculations.


Introduction VTC method VTC ORB MC-VTC Conclusion
Jastrow ansatz for electron correlation where τ (R) is a function with permutational symmetry, and Φ is a single (or multi) determinant reference wave function.

Motivations for Jastrow ansatz
Compact description of electron correlation, Speed up the configuration basis-set convergence (R12, F12, etc.), Platform for new numerical techniques (Wavelets, sparse grids, tensor products, etc.).
Too complicated for implementation.
Desire: simple and efficient equations.

VTC method VTC ORB MC-VTC Conclusion
Variational transcorrelated method

Variational equations
Variational ground state energy T: spanned by a set of basis functions τ (R) = α c α U α (R), F: defined by a given primitive orbital basis set and the rank of Φ. Practical equations Can only be used by QMC.The similarity transformation is used to remove the exponential factor which gives rise to a non-Hermitian term [ Ĥ, τ ], and leads to lose of variational bound.In case of small basis sets with only up to two-body correlation terms, the TC-C equation still serves as a good approximation of equation (V.1), as long as Φ is not much worse than Φ HF .

VTC method VTC ORB MC-VTC Conclusion
Variational transcorrelated equation Replace (V.1) with TC-C in the variational equation, we get the variational transcorrelated equation where the non-Hermitian term [ Ĥ, τ ] simply drops out from equation The variational transcorrelated equation can be applied to a single determinant Jastrow ansatz as well as to a multi-determinant one.

VTC method VTC ORB MC-VTC Conclusion
Variational transcorrelated orbital equation

Effective VTC Hamiltonian
Suppose T contains only up to two body terms The effective Hamiltonian H = Ĥ − 1 2 i ( i τ ) 2 contains then only up to three body operators

VTC method VTC ORB MC-VTC Conclusion
Hartree-Fock type orbital equation Those three-body terms of Fµν have a special structure

VTC method VTC ORB MC-VTC Conclusion
A simple (but not efficient) algorithm Slater type primitive orbital basis, Polynomial basis for the correlation factor QMC calculation of the TC-C matrix elements, Gaussian package for Fork operators (fit STO by GTO basis), Numerical integration for effective potentials (L, K ).

VTC method VTC ORB MC-VTC Conclusion
Multi-configuration variational transcorrelated method Motivations for multi-configuration methods Multi-configuration wave functions are necessary for systems with quasi-degenerate ground states (with low lying excited states), dealing with chemical reactions (transition structures, reactive intermediates, excited electronic states, etc.) Multi-configuration self consistent field (MCSCF) method: Optimize both orbitals and CSF coeficients (a combination of HF and CI methods).
Not efficient for the description of dynamic correlation.
Variational transcorrelated method could be a potential candidate for a new MCSCF approach!

VTC method VTC ORB MC-VTC Conclusion
A transcorrelated CI approach As a preliminary test of the variational transcorrelated method on multi-configuration Jastrow ansatz, we try to solve a CI type transcorrelated equation where the matrix elements are (for the moment) calculated by QMC.

Conclusion
We propose a new transcorrelated method for the calculation of electron correlation, which can be applied to a single determinant Jastrow ansatz as well as a multi-determinant one.For the single determinant ansatz, we obtain a Hartree-Fock type self-consistant equation for the optimization of orbitals, and for the multi-determinant ansatz we have tested a CI type equation.We apply the new equations to C 2 molecule, and the results are in very good agreements with those of variational quantum Monte Carlo.
Future works Extend to variational transcorrelated MCSCF method.
Generate a complete Gaussian package of the VTC method.
Incorporate tensor product technique in VTC.
Transcorrelated and variational correlation energies.The variational energies are calculated with VMC for both TC (τ TC ) and energy optimised correlation functions (τ opt ).The single determinant reference function Φ is fixed as the Transcorrelated and variational ground state energies calculated for C 2 .