Graphitic polymer strips with edge states

Projected Hartree–Fock (PHF) theory has a long history in quantum chemistry. PHF is here understood as the variational determination of an N-electron broken symmetry Slater determinant that minimizes the energy of a projected state with the correct quantum numbers. The method was actively pursued for several decades but seems to have been abandoned. We here derive and implement a “variation after projection” PHF theory using techniques different from those previously employed in quantum chemistry. Our PHF methodology has modest mean-field computational cost, yields relatively simple expressions, can be applied to both collinear and non-collinear spin cases, and can be used in conjunction with deliberate symmetry breaking and restoration of other molecular symmetries like complex conjugation and point group. We present several benchmark applications to dissociation curves and singlet-triplet energy splittings, showing that the resulting PHF wavefunctions are of high quality multireference character. We also provide numerical evidence that in the thermodynamic limit, the energy in PHF is not lower than that of broken-symmetry HF, a simple consequence of the lack of size consistency and extensivity of PHF.

Projected Hartree–Fock theory

Publisher Summary This chapter focuses on the spin-projected extended Hartree–Fock (HF) method. The HF method, in its most usual form, applies the restriction for the one-electron functions that (for closed-shell systems) two electrons with opposite spins are put on the same spatial orbital. Almost all the methods used in quantum chemistry are based on the well-known variation theorem. Choosing the type of the trial wave function used in the variation procedure, one must take into account a great number of different factors—for example, the given class of the wave functions must be adequate to describe the most important physical features of the system, contain as many free variational parameters as feasible from the computational point of view, etc. The one-particle approximations-the Hartree–Fock method and its generalizations-are of upmost importance in this respect. This is one of the reasons why just the HF or selfconsistent field (SCF) method is the basic one in quantum chemistry. Many-electron wave function is approximated as a single Slater determinant—such as, an antisymmetrized product of functions each of which depends on coordinates of only one electron. These one-electron functions (orbitals) are then optimized in a variational way. In addition, a number of actual calculations are also done by using the “extended Hartree–Fock” (EHF) method and the related approaches; these clarified the possibilities and limitations of the method.

The Spin-Projected Extended Hartree-Fock Method

Van der Waals interactions between surfaces of biological interest

Projected Hartree-Fock theory (PHF) has a long history in quantum chemistry. PHF is here understood as the variational determination of an N-electron broken symmetry Slater determinant that minimizes the energy of a projected state with the correct quantum numbers. The method was actively pursued for several decades but seems to have been abandoned. We here derive and implement a "variation after projection" PHF theory using techniques different from those previously employed in quantum chemistry. Our PHF methodology has modest mean-field computational cost, yields relatively simple expressions, can be applied to both collinear and non-collinear spin cases, and can be used in conjunction with deliberate symmetry breaking and restoration of other molecular symmetries like complex conjugation and point group. We present several benchmark applications to dissociation curves and singlet-triplet energy splittings, showing that the resulting PHF wavefunctions are of high quality multireference character. We also provide numerical evidence that in the thermodynamic limit, the energy in PHF is not lower than that of broken-symmetry HF, a simple consequence of the lack of size consistency and extensivity of PHF.

Projected Hartree-Fock Theory

The (spin projected) extended Hartree-Fock equations are derived and discussed in the general (“many λi”) case. The derivation is based on the corresponding generalized Brillouin theorem, substituting in its expression the spin projected DODS Slater determinant as a linear combination of determinants. The expressions obtained for the elements of the different matrices ecd (c = a or b, and d = a or b) occurring in the equations are analyzed. The transformation of the equations into the form of a pseudoeigenvalue problem and the LCAO form of the equations are given, too. Finally the relation to the Goddard's GF equations is discussed in detail.

Spin projected extended Hartree--Fock equations

Energy band structure calculations for poly (A-T) and poly (G-C) in the semiempirical SCF LCAO crystal orbital approximation

Generalizing the method of Hirschfelder, Curtiss and Bird for the calculation of the polarizability of molecules, we have calculated both the diagonal and off-diagonal polarizability tensor elements of the nucleotide bases and base pairs. In the course of calculations also the components of the permanent dipole moments have been computed for these systems.



On the basis of the results obtained we have determined also the critical field strengths necessary to break off a base pair and thus to induce the unwinding of the DNA double helix. According to the data obtained, a field strength of the order 108 V/cm can induce the unwinding of DNA if its direction lies in the plane of the base pairs.

The effect of electric field on the electronic structure of DNA. I. Calculation of the polarizability and of the permanent dipole moment for the nucleotide bases and base pairs

A new electron structural mechanism is proposed for interpretations in molecular biology, in addition to the already existing theories of Szent-Gyorgyi (semiconductivity) and Frohlich (long-range coherence). The hypothetical “intermediate-” or “zigzag states” (ZZS) of solids are investigated by the recursion (transfer matrix) method. The physical reality of the ZZS is discussed up to an SCFDODS-type theory and the necessity of additional less approximate investigations is emphasized. The possible role of ZZS in the explanation of: (i) translation in protein synthesis, (ii) energy and charge transfer processes, as well as (iii) initiation of protein formation is outlined.

Are intermediate states responsible for certain specific properties of biological macromolecules

The π-electron distributions, spin densities, and energies of the first triplets of the nucleotide bases, uracil, thymine, cytosine, adenine, and guanine, were investigated in various semiempirical approximations. Results are presented for calculations using the semiempirical form of the closed-shell SCF configuration interaction method, of the different orbitals for different spins (unrestricted Hartree–Fock) treatment, with and without spin projection, and of the Roothaan's open-shell procedure.

G. Biczó

Papers

Spin projected extended Hartree--Fock equations

Energy band structure calculations for poly (A-T) and poly (G-C) in the semiempirical SCF LCAO crystal orbital approximation

The effect of electric field on the electronic structure of DNA. I. Calculation of the polarizability and of the permanent dipole moment for the nucleotide bases and base pairs

Are intermediate states responsible for certain specific properties of biological macromolecules

The first excited triplet states of the nucleotide bases calculated with different semiempirical SCF schemes