Spin-space

The basic physical idea of HF theory is a simple one and can be tied in very nicely with our discussion of the electron density given in Chapter 5. We noted the physical significance of the density function pi(r, 5) p (r, s)drdv gives the chance of finding any electron simultaneously in the spin-space volume elements dr and dr, with the other electrons anywhere in space and with either spin, / (r) dr gives the corresponding chance of finding any electron with either spin in the spatial volume element dr. 

The Hamiltonian (3.4) is a function of the usual spatial coordinates x, y, z or r, 0, (j)). Electrons possess the intrinsic property of spin, however, which is to be thought of as a property in an independent, or orthogonal, space (spin space). Spin is actually a consequence of the theory of relativity but we shall merely graft on the property in an ad hoc fashion. The spin, s, of an electron (don t confuse with s orbitals ) takes the value 1/2 only. The z component of spin, m, takes (25 + 1) values of ms, ranging 5, 5-l,...-s. Thus for the single electron, = +1/2 or -1/2, also labelled a or p, or indicated by t or i. 

Two important facts concerning the set of relations given above are that all the A -representability relations known to us, can be derived from (45) (or (44) in a spin-space representation) by varying the value of N and relation (49) condenses them all. 

Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd. As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant. 

This quantity is of great importance, since it actually contains all information about electron correlation, as we will see presently. Like the density, the pair density is also a non-negative quantity. It is symmetric in the coordinates and normalized to the total number of non-distinct pairs, i. e., N(N-l).8 Obviously, if electrons were identical, classical particles that do not interact at all, such as for example billiard balls of one color, the probability of finding one electron at a particular point of coordinate-spin space would be completely independent of the position and spin of the second electron. Since in our model we view electrons as idealized mass points with no volume, this would even include the possibility that both electrons are simultaneously found in the same volume element. In this case the pair density would reduce to a simple product of the individual probabilities, i.e.,... 

Here, f(xj x2) is sometimes called the correlation factor. Consequently, IT x, x2) =0 defines the completely uncorrelated case. However, note that in this case, i. e., for f(x2 x2) = 0, p2(x1 x2) is normalized to the wrong number of pairs, since JJ p2llconditional probability l(x2 x1). This is the probability of finding any electron at position 2 in coordinate-spin space if there is one already known to be at position 1... 

Fig. 5 Symmetry-based dipolar recoupling illustrated in terms of pulse sequences for the CN (a) and RNvn (b) pulse sequences, a spin-space selection diagram for the Cl symmetry (c) (reproduced from [118] with permission). Application of POST-CVj [31] as an element in a H- H double-quantum vs 13C chemical shift correlation experiment (d) used as elements (B panel) in a study of water binding to polycrystalline proteins (reproduced from [119] with permission)...

This leads to an exchange-correlation potential in the form of a 2 x 2 Hermitian matrix in spin space... 

Recapitulating, the SBM theory is based on two fundamental assumptions. The first one is that the electron relaxation (which is a motion in the electron spin space) is uncorrelated with molecular reorientation (which is a spatial motion infiuencing the dipole coupling). The second assumption is that the electron spin system is dominated hy the electronic Zeeman interaction. Other interactions lead to relaxation, which can be described in terms of the longitudinal and transverse relaxation times Tie and T g. This point will be elaborated on later. In this sense, one can call the modified Solomon Bloembergen equations a Zeeman-limit theory. The validity of both the above assumptions is questionable in many cases of practical importance. 

The dimensions of the spin spaces for the active electrons in Table 2, cf. Eq. (9)) are certainly not small. It proved difficult to find a spin basis in which very few of the coefficients were large and so we adopted instead a spin correlation scheme cf. Section 4.2). In the present work, we exploited the way in which expectation values of the two-electron spin operator evaluated over the total spin eigenfunction 4, depend on the coupling of the individual spins associated with orbitals ( )/ and j. Negative values indicate singlet character and positive values triplet character. Special cases of the expectation value are ... 

For a two-electron system in 2m-dimensional spin-space orbital, with and denoting the fermionic annihilation and creation operators of single-particle states and 0) representing the vacuum state, a pure two-electron state ) can be written [57]... 

According to the Dirac [36] electron theory, the relativistic wavefunction has four components in spin-space. With the Hermitian adjoint wave function , the quantum mechanical forms of the charge and current densities become [31,40]... 

Finally it will be remarked that the introduction of spin causes no new difficulty, for the spin space is of finite dimension and in consequence quite trivial from our point of view. 

The decomposition eq. (2-6) of the spin-free space FSP induces a decomposition on the Pauli-allowed portion of the Hilbert space of the Hamiltonian H of eq. (2-1). The Hamiltonian H which includes spin interactions may operate on any ket of the space Fsp V", where V is the electronic spin space. Here the symbol indicates a tensor product, so that Fsp Va consists of all spatial-spin kets which are composed of linear combinations of a simple product of a spatial ket of FSP and a spin ket of Va. The Pauli-allowed portion of the total A-electron Hilbert space of is... 

When we take the spin interaction perturbation Q into account, and spin is not a good quantum number, it becomes necessary to invoke the spin space V . 

Ga is a rotation by the angle a in the spin space Va.) There is a two-to-one correspondence from SF El J a d to or SF. This inner subdirect product group is115>117 i7a a Symmetry group of the Hamiltonian H(Qeq),... 

Let us now consider a first quantization operator, hc, that only works in the spin space, so Eq. (5.20 holds. The second quantization representation, h, can be written... 

Three operators S+, S, and Sz works only in the spin space. From the actions of these operators,... 

An irreducible tensor operator of rank S in spin space is defined as a set of 2S+1 operators, T(S,M), with M running from -S to S fulfilling the relations... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

Let us now look at one-particle operators in the second-quantization representation, defined by (13.22). Substituting into (13.22) the one-electron matrix element and applying the Wigner-Eckart theorem (5.15) in orbital and spin spaces, we obtain by summation over the projections... 

Specifically, the operators of kinetic and potential energy of electrons (see (1.15)) are scalars in orbital and spin spaces. Therefore, their sum is at once found to be for a shell nlN... 

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