Symmetric two-electron operators

Before deriving equations that determine the RDMCs, we ought to clarify precisely which are the RDMCs of interest. It is clear, from Eqs. (25a) and (25b), that Ai and A2 contain the same information as D2 and can therefore be used to calculate expectation values (IT), where W is any symmetric two-electron operator of the form given in Eq. (1). Whereas the 2-RDM contains all of the information available from the 1-RDM, and affords the value of (IT) with no additional information, the 2-RDMC in general does not determine the 1-RDM [43, 65], so both Ai and A2 must be determined independently in order to calculate (IT). More generally, Ai,...,A are all independent quantities, whereas the RDMs Dj,..., D are related by the partial trace operation. The u-RDM determines all of the lower-order RDMs and lower-order RDMCs, but... 

The symmetric two-electron operator V is given in terms of the electron-electron potentials by... 

Equn. (5.80) can be formed into a set of equations to be solved for pk Ei — Ei + V i) in analogy to (4.101,4.116), but a close approximation is given by the first iteration, which we write using the second-quantised form (3.149) of the symmetric two-electron operator V as... 

Since H is symmetric F<2) jg also a symmetric two-electron operator... 

The case for which F is a totally symmetric two-electron operator i.e., which X = i/ and a = ir is one that often occurs... 

The quantum numbers listed are for the eigenvalues of the total-spin operators S2 and Sz, where the total spin S is defined as S = S, + S2. Since electrons are fermions, the symmetric two-electron spatial function (1.249) must be multiplied by the antisymmetric spin function (1.254) to give an overall wave function that is antisymmetric the antisymmetric spatial function (1.250) must be multiplied by one of the symmetric spin functions (1.251H1.253). 

Since the many-electron Hamiltonian, the effective one-electron Hamiltonian obtained in Hartree-Fock theory (the Fock operator ), and the one- and two-electron operators that comprise the Hamiltonian are all totally symmetric, this selection rule is extremely powerful and useful. It can be generalized by noting that any operator can be written in terms of symmetry-adapted operators ... 

At this point we have developed a method for handling symmetric one-electron operators a useful step, but far from adequate to cover all cases we are interested in, since we have said nothing yet about the much more demanding case of two-electron integrals. Let us consider first building the Fock matrix for a closed-shell SCF calculation. This is perhaps most easily expressed in terms of a P supermatrix with elements... 

Hamiltonians do not have diagonal matrix elements in a basis of real (Cartesian) orbitals. Further, the use of two-electron operators of the forms [103] or [105] requires a symmetrization in the particle indices... 

Let a Slater determinant , built up from the spin-orbitals totally symmetric hermitean two-electron operator 0(1,2). We wish to determine pair functions of the type (if no summation sign is used, the Einstein summation convention over repeated indices is implied)... 

The Hamiltonian operator is a sum of one- and two-electron operators which is symmetrical in the coordinates of all the particles ... 

Let us now focus on the physical mechanism for charge transfer that emerges from our approach. First, equation (12) predicts a zero probability for the electron exchange H(ls)H H" H(ls) as long as one excludes the electron-electron interaction. We can now introduce the latter effect by approximating the two-electron operator as the symmetrized scalar products that couple the electric field of one subsystem with the dipole moment operator of the other subsystem ... 

The parallels between the one-electron operator and the NMFW one-electron matrices in (19.33) are obvious. It turns out that insertion of the resolution of the identity in a nonorthogonal basis does not strictly lead to the NMFW matrices. These matrices are rather to be regarded as the result of a symmetric reorthonormalization. However, given the approximation to the two-electron operator, the difference betwen the two is not likely to be serious. Armed with the NMFW one-electron matrices, we can write down the NMFW SCF equation. 

The two-electron operator is written so that it is symmetric with respect to permutation of the particle indices. This symmetrization is necessary since we know only how to generate the second-quantization representation of two-electron operators that are symmetric in the particles. 

Figure 2(b) represents the potential surface of the identical system, mapped onto the double-cover space [28], The latter is obtained simply by unwinding the encirclement angle < ), from 0 2ti to 0 4ti, such that two (internal) rotations around the Cl are represented as one in the page. The potential is therefore symmetric under the operation Rin defined as an internal rotation by 2n in the double space. To map back onto the single space, one cuts out a 271-wide sector from the double space. This is taken to be the 0 2ti sector in Fig. 2(b), but any 27i-wide sector would be acceptable. Which particular sector has been taken is represented by a cut line in the single space, so in Fig. 2(b) the cut line passes between < ) = 0 and 2n. Since the single space is the physical space, any observable obtained from the total (electronic + nuclear) wave function in this space must be independent of the position of the cut line. 

The problem of evaluating the effect of the perturbation created by the ligands thus reduces to the solution of the secular determinant with matrix elements of the type rp[ lICT (pk, where rpj) and cpk) identify the eigenfunctions of the free ion. Since cpt) and cpk) are spherically symmetric, and can be expressed in terms of spherical harmonics, the potential is expanded in terms of spherical harmonics to fully exploit the symmetry of the system in evaluating these matrix elements. In detail, two different formalisms have been developed in the past to deal with the calculation of matrix elements of Equation 1.13 [2, 3]. Since t/CF is the sum of one-electron operators, while cpi) and cpk) are many-electron functions, both the formalisms require decomposition of free ion terms in linear combinations of monoelectronic functions. 

The annihilation operators are written to the right of the creation operators to ensure that g operating on an occupation number vector with less than two electrons vanishes. Using that the annihilation operators anticommute and that the creation operators anticommute it is easy to show that the parameters g can be chosen in a symmetric fashion... 

It is also possible to generalize the discussion we have given of the two-electron integral case to operators that are not necessarily totally symmetric, such as those arising in various fine-structure integrals (operators such as f°r example). 

Now, from our manipulations above we can easily see that F can be constructed from a skeleton matrix obtained using only a P2 list, since the Fock operator is totally symmetric. However, this far from ideal, since we would like to avoid the redundancies that arise unless we use the Pi list of two-electron integrals, or, here, supermatrix elements. We define first a matrix Y(IJKL) with the property that the IJ block is given by... 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are... 

Low-symmetry LF operators are time-even one-electron operators that are non-totally symmetric in orbit space. They thus have quasi-spin K = 1, implying that the only allowed matrix elements are between 2P and 2D (Cf. Eq. 28). Interestingly in complexes with a trigonal or tetragonal symmetry axis a further selection rule based on the angular momentum theory of the shell is retained. Indeed in such complexes two -orbitals will remain degenerate. This indicates that the intra-t2g part of the LF hamiltonian has pseudo-cylindrical D h symmetry. As a result the 2S+1L terms are resolved into pseudo-cylindrical 2S+1 A levels (/l = 0,1,..., L ). It is convenient to orient the z axis of quantization along the principal axis of revolution. In this way each A level comprises the ML = A components of the L manifold. In a pseudo-cylindrical field only levels with equal A are allowed to interact, in accordance with the pseudo-cylindrical selection rule ... 

It is also possible to derive this result by incorporating the condition (2.5) below from the beginning by using the factorization of a two-electron function into a symmetrical spatial function and an antisymmetrical spin function, see equ. (1.16).) The expression in the braces indicates that the two electrons in the final state have opposite spins, i.e., the photoprocess reaches a singlet final state. This can be easily understood, because in LS-coupling spin-orbit effects are absent, and the photon operator does not act on the spin. Therefore, the selection rule... 

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