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self-consistent equations of this type, even in their simplified Hartree version, and numerical methods using successive approximations are commonly used. We start from a series of plausible functions , and compute with (46) the associated potentials. Considering them to be fixed, we obtain linear eigenvalue equations which can be solved quite readily with computers (the single very complicated equation in a 3N-dimensional space has been replaced by N independent 3-dimensional equations); we have to diagonalize a Hermitian operator to get a new series of orthonormal functions, resulting from the first iteration, and called and . The second iteration starts from these , to compute the new potential values, and get new linear differential equations. Solving these equations yields the next order , , etc. After a few iterations, one expects the and to vary only slightly with the iteration order (n), in which case the Hartree-Fock equations have been solved to a good approximation. Using (44) we can then compute the energy we were looking for. It is also possible that physical arguments can help us choose directly adequate trial functions φn(r) without any iteration. Inserting them in (44) then directly provides the energy.

       Comments:

      (i) The solutions of the Hartree-Fock equations may not be unique. Using the iteration process described above, one can easily wind up with different solutions, depending on the initial choice for the functions. This multiplicity of solutions is actually one of the method’s advantages, as it can help us find not only the ground level but also the excited levels.

      (ii) As we shall see in § 2, taking into account the 1/2 spin of the electrons in an atom does not bring major complications to the Hartree-Fock equations. It is generally assumed that the one-body potential is diagonal in a basis of the two spin states, labeled + and –, and that the interaction potential does not act on the spins. We then simply assemble N₊ equations, for N₊ wave functions associated with the spin + particles, with N_– other equations, for N_– wave functions associated with spin — particles. These two sets of equations are not independent, since they contain the same direct potential (computed using (46), whose first line includes a summation over p of all the N = N₊ + N_wave functions). As for the exchange potential, it does not lead to any coupling between the two sets of equations: in the second line of (46), the summation over p only includes particles in the same spin state for the following reason. If the particles have opposite spins, they can be recognized by the direction of their spin (the interaction does not act on the spins), and they no longer behave as indistinguishable particles. The exchange effects only arise for particles having the same spin.

2. Generalization: operator method

      We now describe the method in a more general way, using an operator method that leads to more concise expressions, while taking into account explicitly the possible existence of a spin – which plays an essential role in the atomic structure. We will identify more precisely the mathematical object, actually a projector, which we vary to optimize the energy. Physically, this projector is simply the one-particle density operator defined in § B-4 of Chapter XV. This will lead to expressions both more compact and general for the Hartree-Fock equations. They contain a Hartree-Fock operator acting on a single particle, as if it were alone, but which includes a potential operator defined by a partial trace which reflects the interactions with the other particles in the mean field approximation. Thanks to this operator we can get an approximate value of the entire system energy, computing only individual energies; these energies are obtained with calculations similar to the one used for a single particle placed in a mean field. With this approach, we have a better understanding of the way the mean field approximately represents the interaction with all the other particles; this approach can also suggest ways to make the approximations more precise.

      We assume as before that the N-particle variational ket is written as:

(47)

This ket is derived from N individual orthonormal kets |θk〉, but these kets can now describe particles having an arbitrary spin. Consider the orthonormal basis {|θk〉} of the one-particle state space, in which the set of |θi〉 (i = 1, 2, …N) was completed by other orthonormal states. The projector PN onto the subspace is the sum of the projections onto the first N kets |θi〉:

(48)

      This is simply the one-particle density operator defined in § B-4 of Chapter XV (normalized by a trace equal to the particle number N and not to one), as we now show. Relation (B-24) of that chapter can be written in the |θk〉 basis:

(49)

where the average value is taken in the quantum state (47). In this kind of Fock state, the average value is different from zero only when the creation operator reconstructs the population destroyed by the annihilation operator, hence if k = l, in which case it is equal to the population nk of the individual states |θk〉. In the variational ket (47), all the populations are zero except for the first N states |θi〉 (i = 1, 2, …N), where they are equal to one. Consequently, the one-particle density operator is represented by a matrix, diagonal in the basis |θi〉, and whose N first elements on the diagonal are all equal to one. It is indeed the matrix associated with the projector PN, and we can write:

(50)

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