One-particle transformations

Consider a set of creation and annihilation operators a and aj. The most general linear transformation of these operators can be written as  

Assume that the original operators a, a obey the usual fermion anticommutation rules  

The transformation of Eq. (16.1) is said to be canonical if the same rules are maintained for the transformed operators  

These equations give some conditions for the transformation coefficients A and B which can be obtained by substituting Eq. (16.1) into Eq. (16.2)  

These are the conditions for the quasiparticle transformation to be a canonical transformation. Some special cases of such transformations we have seen previously. If, for example, Bij, = 0, one has  

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The condition for E to be stationary with respect to unitary one-particle transformations in generalized normal order is... 

In conclusion, it should be observed that in many applications it is possible to express the AT-particle transformation U in terms of one-particle transformations u, e.g., in the product form... 

Let us now consider the one-particle transformations (3.19) which, in this case, take the form ... 

In order to establish the transformation properties of the one-particle states, let us obtain the transformation properties of the in operators. 

This is verified by applying U(a,A) to the left and U(a,A) l to the right of both sides of Eqs. (11-142) and (11-97), mid making use of the transformation properties of (11-184) and (11-185) of the fields. Equations (11-241) and (11-242) were to be expected. They guarantee that four-vector, respectively. We are now in a position to discuss the transformation properties of the one-particle states. Consider the one-negaton state, p,s,—e>. Upon taking the adjoint of Eq. (11-241) and multiplying by ya we obtain... 

This expression is derived as the Fourier transform of a time-dependent one-particle autocorrelation function (26) (i.e. propagator), and cast in matrix form G(co) over a suitable molecular orbital (e.g. HF) basis, by means of the related set of one-electron creation (ai" ") and annihilation (aj) operators. In this equation, the sums over m and p run over all the states of the (N-1)- and (N+l)-electron system, l P > and I P " respectively. Eq and e[ represent the energy of the... 

This one-particle equation is sufficiently simple so that it is possible to obtain numerical solutions to any degree of accuracy. As first done by Burrau [84] the equation is transformed (eqn. 1.12) into confocal elliptic coordinates... 

Density-Transformed One-Particle Orbitals and Their use in the Construction of Energy Functionak... 

All transformed wavefunctions yield the exact (Benesch [68]) one-particle density. 

Let us consider an N-particle wavefunction pj( i,..., r y) (where for simplicity we disregard spin) associated with the one-particle density Pi(r), and let us obtain the Fourier transform of this wavefunction by means of ... 

Local-scaling transformations have been employed [39] in order to obtain a one-particle density in position space from the one-particle density in momentum space, and vice versa. This problem arises from example when we have a y(p), obtained from experimental Compton profiles, and wish to calculate the corresponding p(r) [98]. 

Let us now consider the transformed orbitals from the perspective of localscaling transformations. For a fixed set of single-particle functions <, (r) and a fixed set of expansion coefficients Cr, the wavefunction given by Eq. (114) is also fixed and yields a one-particle density p (r). We now consider the localscaling transformation that carries this density into a density p(r) and obtain the corresponding transformed orbitals ... 

The uniqueness of the local-scaling transformation guarantees that within an orbit there exists a one-to-one correspondence between one particle... 

Moreover, the construction of functionals for the energy that depend explicitly on the one-particle density is, of course, quite feasible, if one uses analytic approximations for the transformed vector/([p] r). Such an alternative is open if, for instance, one resorts to the use of Fade approximants. We discuss this way of dealing with this problem in Sect. 5. 

It follows, therefore, that the transformed Slater determinant also depends upon the parameters a , Cnsp that define the transformed orbitals. Moreover, if we select the following representation for the one-particle density ... 

We review in this Section some recent work by Ludena, Lopez-Boada and Pino [113] on the construction of energy functionals that depend explicitly upon the one-particle density, but which are generated in the context of the local-scaling-transformation version of density functional theory. This work does not consider the general case involving exchange and correlation, but restricts itself to the exchange-only Hartree-Fock approximation. 

For the Hartree-Fock case, the energy functional [4> >] for the density-transformed single Slater determinant given by Eq. (152), may be rewritten as a functional of the one-particle density ... 

As discussed above, this still remains as an implicit functional of the one-particle density, in view of the fact that/(r) = /([p] r). Nevertheless, when we introduce the analytic Fade approximant for/([p] r) given by Eq. (189), we can transform Eq. (200) into an explicit functional of p. 

While all three matrices are interconvertible, the nonnegativity of the eigenvalues of one matrix does not imply the nonnegativity of the eigenvalues of the other matrices, and hence the restrictions Q>0 and > 0 provide two important 7/-representability conditions in addition to > 0. These conditions physically restrict the probability distributions for two particles, two holes, and one particle and one hole to be nonnegative with respect to all unitary transformations of the two-particle basis set. Collectively, the three restrictions are known as the 2-positivity conditions [17]. 

Physically, the 3-positivity conditions restrict the probability distributions for three particles, two particles and one hole, one particle and two holes, and three holes to be nonnegative with respect to all unitary transformations of the one-particle basis set. These conditions have been examined in variational 2-RDM calculations on spin systems in the work of Erdahl and Jin [16], Mazziotti and Erdahl [17], and Hammond and Mazziotti [33], where they give highly accurate energies and 2-RDMs. 

Thus we see that Hartree-Fock theory is identical to a canonical transformation theory retaining only one-particle operators with an optimized reference, and the canonical transformation model retaining one- and two-particle operators employed in the current work, if employed with an optimized reference, is a natural extension of Hartree-Fock theory to a two-particle theory of correlation. 

Given a wavefunction built from a one-particle basis set, it is convenient [9,197,198] to proceed with the computation of the six-dimensional Fourier transform in Eq. (5.15) by inserting the spectral expansion [123,126]... 

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