Pair Functionals

At the next level of sophistication beyond pair potentials, a variety of methods which can broadly be classified as pair functionals have been developed. In keeping with the various surrogates for the full Hamiltonian introduced in eqns (4.4), (4.5) and (4.30), the total energy in the pair functional setting is given by 

The philosophy of the pair functional approach is perhaps best illustrated via a concrete example. Consider fee Ni. In this case, we know on the basis of the results shown above, that the Ni atom prefers to live in a certain equilibrium electron density. As yet, we have not seen how to exploit this insight in the context of particular geometric arrangements of a series of Ni atoms. We may see our way through this difficulty as follows. If we center our attention on a particular Ni atom, we may think of it as being deposited in an electron gas, the density of which is dictated by the number and proximity of its neighbors. The hypothesized energy 

This equation tells us how to determine the density of the electron gas at the site of interest which we denote pi in terms of the geometric disposition of the neighboring atoms. In particular, f(R) is a pairwise function that characterizes the decay of electronic density from a given site. Hence, if we wish to find the contribution of atom j to the density at site i, the function f(R) evaluated at the distance R = Rj — R tells us exactly how much electronic density bleeds off from site j onto its neighbors. 

For the linear chain of atoms (see fig. 4.7), the total energy per atom within the embedded atom context is given by 

This result reflects the fact that every atom on the chain has an identical environment with two near neighbors at the equilibrium distance. If we continue along the lines set above and consider the total energy of the square lattice depicted in fig. 4.7, it is seen to have a total energy of 

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Again, a closure is needed. Even with a closure, the system of equations is not complete. A relation between the singlet function p(r) and the pair functions is needed. For this purpose the first equation of the BGY hierarchy may be used. Alternatively, one can apply the Lovett-Mou-Buff-Wertheim equation [100,101]... 

Sokolowski [102,103] and Phschke and Henderson [104,105] have apphed the lOZ equation, with the PY and HNC closures, to hard spheres near a hard wall. Using the HNC and MSA closures, Plischke and Henderson [106-108] have studied charged hard spheres near a charged planar hard wall. The results are very good. In contrast to the results obtained from the OZ equation, the lOZ yields results for p(r) that are nearly independent of the closure. As might be expected, the results for the pair functions are closure dependent and show similar problems to those seen for the RDF with the OZ and similar closures. 

Finally, the closure relations for the inhomogeneous pair functions must be chosen. The PY approximation for the fluid-fluid direct correlation function presumes that its blocking part vanishes. This implies that c, ii(/,y) = 0, and... 

Coupled—Pair Functional (ACPF) and Coupled Electron Pair Approximation (CEPA). The simplest form of CEPA, CEPA-0, is also known as Linear Coupled Cluster Doubles (LCCD). 

Figure4.7 Relativistic bond contractions A re for Au2 calculated in the years from 1989 to 2001 using different quantum chemical methods. Electron correlation effects Acte = te(corn) — /"e(HF) at the relativistic level are shown on the right hand side of each bar if available. From the left to the right in chronological order Hartree-Fock-Slater results from Ziegler et al. [147] AIMP coupled pair functional results from Stbmberg and Wahlgren [148] EC-ARPP results from Schwerdtfeger [5] EDA results from Haberlen and Rdsch [149] Dirac-Fock-Slater...

Kofraneck and coworkers24 have used the geometries and harmonic force constants calculated for tram- and gauche-butadiene and for traws-hexatriene, using the ACPF (Average Coupled Pair Functional) method to include electron correlation, to compute scaled force fields and vibrational frequencies for trans-polyenes up to 18 carbon atoms and for the infinite chain. 

In contrast, the NBO picture describes (aAB)2 in terms of a doubly occupied bond orbital pair function t// ab(NB0) (cf. Eq. (1.31a)), with spatial dependence (in the closed-shell singlet case)... 

As a characteristic feature, both the gap functions have nodes at poles (9 = 0,7r) and take the maximal values at the vicinity of equator (9 = 7t/2), keeping the relation, A > A+. This feature is very similar to 3P pairing in liquid 3He or nuclear matter [17, 18] actually we can see our pairing function Eq. (39) to exhibit an effective P wave nature by a genuine relativistic effect by the Dirac spinors. Accordingly the quasi-particle distribution is diffused (see Fig. 3)... 

The symmetry requirements and the need to very effectively describe the correlation effects have been the main motivations that have turned our attention to explicitly correlated Gaussian functions as the choice for the basis set in the atomic and molecular non-BO calculations. These functions have been used previously in Born-Oppenheimer calculations to describe the electron correlation in molecular systems using the perturbation theory approach [35 2], While in those calculations, Gaussian pair functions (geminals), each dependent only on a single interelectron distance in the exponential factor, exp( pr ), were used, in the non-BO calculations each basis function needs to depend on distances between aU pairs of particles forming the system. 

In this form, the w-particle correlated Gaussian is a product of n orbital Gaussians centered at the origin of the coordinate system and n n — l)/2 Gaussian pair functions (geminals). 

The synthesis of a polyfunctional molecule can always be reduced to the problem of constructing differently paired functional group relationships, which (keeping in mind generalisation 1) usually requires the creation of a carbon chain with a number of carbon atoms equal to or smaller than 6 (n < 6). 

There are useful two- and many-electron analogues of the functions discussed above, but when the Hamiltonian contains only one- and two-body operators it is sufficient to consider the pair functions thus the analogue of p(x x ) is the pair density matrix 7t(xi,X2 x i,x ) while that of which follows on identifying and integrating over spin variables as in (4), is H(ri,r2 r i,r2)- When the electron-electron interaction is purely coulombic, only the diagonal element H(ri,r2) is required and the expression for the total interaction energy becomes... 

The Na+/proline transporter of E. coli (PutP) is an integral membrane protein that is proposed to contain 13 transmembrane helices.85 Four-pulse DEER measurements found distances of 48,22, and 18 A for three doubly spin-labelled variants.The 48 A distance confirmed that those two labels were on opposite sides of the membrane. The large distance distribution widths that were observed in the pair functions reveal the substantial flexibility of the loop regions to which the spin labels were attached. 

Exercise. The basic reason why the fn are more useful than the Qn for describing dots is the following. Most quantities A in whose average one is interested are sum functions , i.e., they consist of a single-particle function a(xa) summed over all particles, or of a pair function fl(tff,v) summed over all pairs etc. In general,... 

Another point worth making is that since the SD-CI method is exact within the chosen basis set for a two-electron system, it must be size-consistent in this particular case. Nevertheless, when Davidson s correction is applied to an SD-CI wave-function for a two-electron system it will give a non-zero contribution, which is thus an artefact of this correction. (The same error appears also when the functional (10.1) is used with g=0.) This artefact can be simply removed and this is done in the Averaged Coupled Pair Functional (ACPF) method. In this method the factor g is considered to be a function of the number of electrons N, g=g(N), and one considers the special case of n separated He atoms. If the denominator De in (10.1) for one He atom is... 

R.J. Gdanitz and R. Ahlrichs, The Averaged Coupled-Pair Functional (ACPF) A Size-Extensive Modification of MR CI(SD), Chem. Phys. Letters 143, 413 (1988). 

Very similar in spirit to CEPA, but formulated as a functional to be made stationary, is the coupled-pair functional (CPF) approach of Ahlrichs and co-workers [28]. CPF can be viewed as modifying the CISD energy functional to obtain size-extensivity for the special case of noninteracting two-electron systems. One disadvantage of some of the CEPA methods is that, unlike CISD or CCSD, the results are not invariant to a unitary transformation that mixes occupied orbitals with one another. CPF... 

Gdanitz and Ahlrichs devised a simpler variant of CPF, the averaged coupled-pair functional (ACPF) approach [30]. This produces results very similar to CPF for well-behaved closed-shell cases and is completely invariant to a unitary transformation on the occupied MOs. Its big advantage is that it can be cast in a multireference form. Multireference ACPF is probably the most sophisticated treatment of the correlation problem currently available that can be applied fairly widely, although it can encounter difficulties with the selection of reference spaces, as discussed elsewhere. 

In Fig. 4, the function tp (Y)/A is plotted against Yq2 for the correct quasielastic pair function and for the pre-average approximation, where the free draining term (kT/ j) djk has been neglected. The deviations between the two curves can be approximated by82 (see Fig. 4). 

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