Professor Fulde's primary purpose in this advanced monograph, as stated in the preface, is to provide a unified treatment of electron correlations in molecules and in "solids"; by the latter, he means materials such as metals in which the electrons are not well localized. The basic idea is that since the electron correlation hole is a local object, it should be possible to develop a theory that applies equally well to electron correlations in both cases. First, let us say what is meant by electron correlation. One starts with a Hamiltonian H that describes an atomic system, say, with n electrons; H acts on a Hilbert space S. Pick a finite-dimensional subspace S_0 and apply a variational principle to minimize the expectation E = <\Phi, H \Phi> over all wavefunctions \Phi expressible as a single Slater determinant of n single-electron wavefunctions \phi_i, 1 <= i <= n. The latter can be any normalized vector in S_0; we form the Slater determinant and minimize E over all such choices of the \phi_i. This leads to the Hartree--Fock equations (2.3.13). In the limit as S_0 increases to all of S, E will decrease to a value E_{HF} called the Hartree--Fock limit. It will be greater than the true ground-state energy E_0, because the true ground-state wavefunction is a convergent sum of Slater determinants, not a single determinant. The difference E_{HF} - E_0 has an interesting physical interpretation that justifies calling it the electron correlation energy. This follows from examining the Coulomb term in the one-particle Fock equations (2.3.13) -- each electron is treated as though it interacted with the average charge distribution of the remaining electrons, whereas in fact it interacts with the actual distribution. The effect of this is to overestimate the ground state energy by allowing electrons to come too close, closer than the true Coulomb force would allow. Thus, if we haven't left out necessary terms in the Hamiltonian (such as spin-orbit coupling, relativistic effects for heavy transition metals, etc.), it is natural to attribute the difference E_{HF} - E_0 to this "electron correlation" effect. The correlation suppresses the likelihood of other electrons being close to a given one as compared to what the HF equations would predict, hence the term "correlation hole" and its local nature. Electron correlation effects are of supreme importance in chemistry, because although generally small compared to the self-consistent field energies E_{SCF}, when chemical bonding occurs the difference in the correlation energy corrections is typically not small compared to the difference in the E_{SCF}. Furthermore, the correlation effects are expected to be exquisitely sensitive to the geometry of the orbitals, so we have a fascinating problem lying at the heart of chemistry. The first four chapters give a rapid, somewhat demanding, and authoritative overview of the basic approximation schemes (the one described above is called the independent electron approximation). The important topics, including density functional theory, are treated. After this promising start, in Chap. 5 Prof. Fulde takes up the main business of the book -- developing a general scheme for constructing localized basis functions optimized so as to account for electron correlation. If he had succeeded in this, it would be a major contribution to the subject. But alas, I have to report that due to a disastrous mathematical oversight, his results are invalid. The smoking gun is Eq. (5.1.20); this does NOT give a well- defined inner product because it is not bilinear and cannot be made so in a consistent, well-defined way. The root of the trouble is the cumulant formula (5.1.15). Let's call the A_i there "atoms" (with respect to cumulant formation). The problem is Fulde's tacit assumption that if the operator Op is a sum of monomials in the A_i then <\Phi_0|Op|\Phi_0>^c can be defined as the corresponding sum of the cumulants of the individual monomials. But what if this expression as a sum of monomials isn't unique? If we write Op in a different way, does the prescription give the same result? The answer is no! Cumulant formation does not respect the ordinary rules of algebra; in particular, one cannot substitute. (I give two examples below.) But in (5.1.20), A and B can be expressed in many different ways as a sum of monomials in the cumulant atoms (here H_1 and possibly H_0). Without a well-defined -- bilinear, positive-semidefinite -- inner product, the entire treatment collapses. This seems to explain the otherwise puzzling lack of references in the literature to the work of Fulde's school; for example, there are none in Vol. 93 in the Advances in Chem. Phys. series (Prigogine, Rice eds.), entitled New Methods in Computational Quantum Mechanics, nor in many other places I have looked. Here are my examples. A is an atom for cumulant formation, and I use only the formulas ^c = and ^c = - ^2. 1) Let A be the two-by-two matrix with 0, 0 in the first column and 1, 0 in the second; with the standard unit vectors e_1 and e_2, we have A e_1=0, A e_2=e_1, and A^2 = 0. Set e = (e_1 + e_2)/sqrt(2) and define the expectation by E(Op) = . Then 0 = <0>^c is not equal to ^c = -^2=0-(1/2)^2=-1/4. Here we have expressed zero in two different ways: as the scalar 0 and as the operator A^2, and have gotten different results. 2) Take A = diag(x, y) with x and y real roots of a polynomial \Sum_0^n c_k w^k with scalar coefficients c_k. Set E(Op) = . Then 0 = \Sum_0^n c_k A^k as an operator, but \Sum_0^n c_k ^c does not equal zero. Take, e.g., n=2, x=2, y=3; they satisfy w^2-5w+6=0. We have \Sum_0^2 c_k ^c = 6<1>^c - 5^c + 1^c = 6 - 5 x 2 +-^2 = 6-10+4-4=-4. So, let readers of this book be warned! |