Antisymmetrized Geminal Power

As in general all the y-coefficients do not vanish one has to assume a more general reference state than the single determinant SCF state. This is the rather well-known problem of finding the consistent reference state for the Random Phase Approximation (RPA). It also means that the field operator basis can be enlarged and can for instance include the iV-electron occupation number operators (in this discussion, electron field operators and their adjoints are used referring to a basis of spin orbitals that are the natural spin orbitals of the reference state, as will be discussed below, i.e., the spin orbitals that diagonalize the one-matrix) 

The metric matrix elements between operators qu and the operators 6 6/ and their adjoints also vanish. 

It has been shown that for an even number of electrons 2N, the appropriate correlated reference state to consider is 

Such a reference state has been called an antisjrmmetrizedgeminal power (AGP). Should the number of electrons be odd 2N + 1, one may construct the appropriate reference state as a generalized antisynunetrized geminal power (GAGP) 

From this expression, one can see that the AGP can be represented as a leading determinant and all double, quadruple, hexatuple, e.t.c., even nonlinked replacements out of that determinant. The mixing coefficients of these determinants 

---

A. J. Coleman, Stmcture of fermion density matrices. 2. Antisymmetrized geminal powers. J. Math. Phys. 6(9), 1425-1431 (1965). 

Some interconnections can be mentioned here. The first concerns Coleman s so-called extreme state (17) (cf. the theories superconductivity and superfluidity). If h is a set of two particle determinants and the wave function is constructed from an antisymmetric geminal power, based on gi, then the reduced density matrix can be expressed as... 

Blatt [36], Coleman [37, 38], then Bratoz and Durand [39] investigated a special N-electron wave function in which all geminals were constrained to have the same form, and established the relationship between this function and that used by Bardeen, Cooper and Shieffer (BCS) [40] to describe superconducting systems with electron pairs. The underlying wave function was termed as the antisymmetrized geminal power (AGP) function. 

The HFB state describes a situation in which the number of particles is not fixed. As shown by Weiner and Goscinski,41 an HFB state can be written as a linear combination of antisymmetrized geminal power (AGP) states for systems with 0, 2, 4, 6,. . . particles. Consequently one can project out of an HFB state the component corresponding to the number of particles of the system under consideration. 

B. Weiner and O. Goscinski Calculation of Optimal Generalized Antisymmetric Geminal Power Functions (projected-BCS) and their Associated Excitation Spectrum Phys. Rev. 22, 2374 (1980). 

B. Weiner and O. Goscinski Excitation Operators Associated with Antisymmetrized Geminal Power (AGP) States Phys. Rev. 27, 57 (1982). 

O. Goscinski and B. Weiner The Excitation Spectrum Associated with a Generahzed Antisymmetrized Geminal Power Ground State Phys Rev. A25, 650 (1982). 

In passing we note that the functions in the set g are completely delocalized over the region of sites defined by the localized particle-antiparticle basis h, while the f-basis contains all possible phase-shifted contributions from each site in accordance with Eqs. (56) and (57) above. Some interrelationships can be recognized here. The first connection concerns Coleman s so-called extreme state [18], cf. the theories of superconductivity and superfluidity based on ODLRO. The second observation relates to the identification of the present finite dimensional representation as a precursor for possible condensations, developing correlations and coherences that may extend over macroscopic dimensions. If h is a set of two-particle determinants and the iV-particle fermionic wave function is constructed from an AGP, antisymmetrized geminal power, based on i, see Eq. (57), then the reduced density matrix can be represented as... 

The centrality of the FNA has spawned considerable research into improvement of the approach. The strategies for obtaining better nodes are numerous. Canonical HF orbitals, Kohn-Sham orbitals from density functional theory (DFT), and natural orbitals from post-HF methods have been used. The latter do not necessarily yield better nodes than single configuration wave functions [39-41]. More success has been found with alternative wave function forms that include correlation more directly than sums of Slater determinants. These include antisymmetrized geminal power functions [42,43], valence-bond [44,45] and Pfaffian [46] forms as well as... 

Historically, MO methods have dominated trial-function construction because these functions are readily obtained from widely distributed computer codes. Recently, however, some QMC practitioners have renewed interest in a broader variety of wave functions including valence bond (VB) functions [44, 45], pairing wave functions, such as the antisymmetrized geminal power (AGP) [42, 43], Pfafflan, and perfect pairing forms [46]. 

An alternative approach to improving upon HF wave functions is to include correlations between pairs of electrons more directly by means of two particle geminal functions, G xi Xj). The antisymmetrized geminal power (AGP), Pfaffian and perfect pairing wave functions are all examples of pairing wave functions each can be written as an antisymmetrized product of geminals,... 

Expansion in terms of determinants is not the only way to represent accurate electronic wave functions. Recently, other types of functions have been employed in QMC calculations with great success. With only one pair function (pg a BCS-type wave function, also known as antisymmetrized geminal power (AGP) is obtained... 

---