The notion of 'Optimal Hartree Fock method' is not familiar to me; perhaps you have meat 'Optimized Effective Potential (OEP) method' [1] within  the framework of the Hartree-Fock approximation.

Be it as it may, one-particle theories of interacting many-particle systems are necessarily approximate in regard to the excitation energies, since they cannot possibly take exact account of the electron-electron interaction effects. To appreciate this, consider the Hartree-Fock method, which is deduced by restricting oneself to a single Slater N-determinant representing a variational N-particle wave function; the self-consistent solutions of the one-particle Hartree-Fock eigenvalue problem merely make the expectation value of the exact N-particle Hamiltonian with respect to the above-mentioned single Slater N-determinant stationary [2]. There is no a priori reason for the eigenvalues of the Hartree-Fock equation to be even approximately equal to the one-particle excitation energies in an interacting system. The same therefore applies to approximations within the framework of the Hartree-Fock theory.

One can view the Hartree-Fock equation in the context of the exact equation for quasi-particles in terms of the self-energy [3], with the self-energy being approximated by the first-order contribution in its perturbation series expansion in powers of the coupling constant of the electron-electron interaction potential. The justification of the Hartree-Fock theory can therefore only be a posteriori, when one can show that the higher-order neglected terms are indeed negligible. To this end, one compares one's calculated results with experiments. Thus, failure of the Hartree-Fock theory to reproduce the experimental one-particle excitation energies signifies that the neglected higher-order terms in the self-energy are not negligible.

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[1] The optimized effective potential method is originally due to Sharp and Horton (1953), and Talman and Shadwick (1976):

RT Sharp, and GK Horton, Phys. Rev. 90, 317 (1953).

JD Talman, and WF Shadwick, Phys. Rev. A14, 36 (1976).

[2] This variational N-particle wave function is constructed from the N lowest-lying eigenfunctions of the self-consistent Hartree-Fock equation.

[3] AL Fetter, and JD Walecka, Quantum Theory of Many-Particle Systems (Dover, New York, 2003).

Well,

Thanks for your nice comments Respected Sir.

but I am not talking about Optimized Effective Potential (OEP) method

but my question is about Optimal Hartree fock (OHF) method

You are welcome.

As I wrote in my first comment on this page, I do not know what OHF is. Perhaps you wish to provide a publicly-available reference in which the method is described. In any case, again as I also wrote in my first comment, no matter what one-electron theory one uses, it can never take account of the excitation energies over an indefinite range -- electron-electron interaction does not lend itself to being described in terms of a fixed static configuration of the electrons. In the quasi-particle formulation, a quasi-particle experiences a different potential as its energy varies, and this possibility falls outside the domain of the Hartree-Fock theory, whether optimized or otherwise.

Thanks for your comments sir, here is link of the paper

http://pubs.acs.org/action/doSearch?text1=Computational+Prediction+for+Singlet-+and+Triplet-Transition+Energies+of+Charge-Transfer+Compounds&quickLinkYear=&quickLinkVolume=&field1=Title&type=within&publication=40026032

You are welcome Aftab.

I asked for a publicly-available reference, that is a reputable publication that anyone can access, where the OHF is described; the flowchart that you present, does not describe the formalism, but tells 'Calculate the optimal HF'. I am very sorry, but I cannot help.