P space

Next, the full-Hilbert space is broken up into two parts—a finite part, designated as the P space, with dimension M, and the complementai y part, the Q space (which is allowed to he of an infinite dimension). The breakup is done according to the following criteria [8-10] ... 

To continue, we define the following two relevant Feshbach projection operators [79], namely. Pm, the projection operator for the P space... 

S u P space. If the determinants j> are built on orthogonal orbitals, equation (6) is automatically fulfilled which ensures that equation (5) is also valid due to the definition of H°. The matrix elements of H° are then easily calculated ... 

Consider a gas whose phase density in T space is represented by a microcanonical ensemble. Let it consist of molecules with //-spaces pi with probability distributions gt. Denote the element of extension in pi by fa. Since energy exchanges may occur between the molecules, pi cannot be represented by a microcanonical distribution. There must be a finite density corresponding to points of the ensemble that do not satisfy the requirement of constant energy. Nevertheless, the simultaneous probability that molecule 1 be within element di of its p-space, molecule 2 within dfa of its //-space, etc., equals the probability that the whole gas be in the element... 

It is noticeable that correlation-consistent basis sets are not able to accurately reproduce the bond distances and energies of the excited states in II symmetry, which may be attributed to the relatively inadequate p-space. The problem was eliminated in the case of H-73 basis set. The polarization set that was purposefully optimized to correlate with the 2p orhitals of hydrogen atoms greatly improved the bond lengths and excitation energies, and reduced the errors to be within 100 cmof exact values. 

The momentum-space orthonormality relation for hydrogenlike Sturmian basis sets, equation) 17), can be shown to be closely related to the orthonormality relation for hyperspherical harmonics in a 4-dimensional space. This relationship follows from the results of Fock [5], who was able to solve the Schrodinger equation for the hydrogen atom in reciprocal space by projecting 3-dimensional p-space onto the surface of a 4-dimensional hypersphere with the mapping ... 

By adopting the no-pair approximation, a natural and straightforward extension of the nonrelativistic open-shell CC theory emerges. The multireference valence-universal Fock-space coupled-cluster approach is employed [25], which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. The effective Hamiltonian has the form [26]... 

Recall that in his Theorems 3 and 4 Hans Kummer [3] defined a contraction operator, L, which maps a linear operator on A-space onto an operator on p-space and an expansion operator, E, which maps an operator on p-space onto an operator on A-space. Note that the contraction and expansion operators are super operators in the sense that they act not on spaces of wavefunctions but on linear spaces consisting of linear operators on wavefunction spaces. If the two-particle reduced Hamiltonian is defined as... 

The counterpart wavefunction in momentum-space, 4>(yi,y2 is a function of momentum-spin coordinates % = (jpk, k) in which pk is the linear momentum of the feth electron. There are three approaches to obtaining the momentum-space wavefunction, two direct and one indirect. The wavefunction can be obtained directly by solving either a differential or an integral equation in momentum- or p space. It can also be obtained indirectly by transformation of the position-space wavefunction. 

The technology for solving the Schrddinger equation is so much farther advanced in r space than in p space that it is most practical to obtain the momentum-space from its position-space counterpart The transformation theories of Dirac [118,119] and Jordan [120,121] provide the hnk between these representations ... 

If the r-space wavefunction is a linear combination of Slater determinants constructed from a set of spin-orbitals /. , then its p-space counterpart is the... 

If we are interested only in properties that can be expressed in terms of q-electron operators, then it is sufficient to work with the th-order reduced-density matrix rather than the A -electron wavefunction [122-126]. In this section, we consider links between the r- and p-space representations of reduced-density matrices. In particular, we show that if we need the th-order density matrix in p space, then it can be obtained from its counterpart in r space without reference to the /-electron wavefunction in p space. 

The p-space density matrices form a hierarchy related through the analog of Eq. (5.7). 

The r-space and p-space representations of the ( th-order density matrices, whether spin-traced or not, are related [127] by a fif -dimensional Fourier transform because the parent wavefunctions are related by a 3A -dimensional Fourier transform. Substitution of Eq. (5.1) in Eq. (5.8), and integration over the momentum variables, leads to the following explicit spin-traced relationship ... 

Then, instead of performing the six-dimensional integral in Eq. (5.19) all at once, we perform successive three-dimensional integrals over s and R. The first step takes us to W R,P), the Wigner representation [130,131] of the density matrix, and the second step to the p-space density matrix, n(P — p/2 P + p/2). The reverse transformation of Eq. (5.20) can also be performed stepwise over P and p to obtain A( , p), the Moyal mixed representation [132], and then the r-space representation V R— s/2 R + s/2). These steps are shown schematically in Figure 5.2. 

Figure 5.2. The relationships among the r-space density matrix F, the p-space density matrix n, the Wigner representation W, and the Moyal representation A. Two-headed arrows with a T beside them signify reversible, three-dimensional, Fourier transformations.

Although the r- and p-space representations of wavefunctions and density matrices are related by Fourier transformation, Eqs. (5.19) and (5.20) show that the densities are not so related. This is easily understood for a one-electron system where the r-space density is just the squared magnitude of the orbital and the p-space density is the squared magnitude of the Fourier transform of the orbital. The operations of Fourier transformation and taking the absolute value squared do not commute, and so the p-space density is not the Fourier transform of its r-space counterpart. In this section, we examine exactly what the Fourier transforms of these densities are. 

B( ) is variously called the reciprocal form factor, the p-space form factor, and the internally folded density. B(s) is the basis of a method for reconstructing momentum densities from experimental data [145,146], and it is useful for the r-space analysis of Compton profiles [147-151]. The B(s) function probably first arose in an examination of the connection between form factors and the electron momentum density [129]. The B f) function has been rediscovered by Howard et al. [152]. 

Figure 5.3. Connections among r- and p-space densities, density matrices, and form factors. Two-headed arrows signify reversible transformations single-barbed arrows signify irreversible transformations. A Fourier transform is denoted by JF.

How is the form factor related to the p-space density matrix Substitution of Eq. (5.20) into Eq. (5.31) and integration over r andp yields [127,129]... 

This equation connects the large-p behavior of the momentum density with the small-r behavior of the electron density and small-w behavior of the intracule density. Hence, Eq. (5.49) is a quantitative manifestation of the reciprocal nature of r and p space. 

Here, the p-space natural momentals 4>i(p) Fourier transforms of the r-space natural orbitals ... 

Thus there is an isomorphism between the first-order, r-space density matrix F and its p-space counterpart II, just as there is an isomorphism between a r-space wavefunction built from a one-particle basis set and the corresponding p-space wavefunction as described in Section II. 

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