Spin-orbital Fock operators

Note that the first and second terms on the right-hand side of this equation are simply the spin-orbital Fock operator (in normal-ordered form), and the last two terms are the Hartree-Fock energy (i.e., the Fermi vacuum expectation value of the Hamiltonian). Thus, we may write... 

To proceed we must evaluate the left-hand side of (3.114). Let us write the spin orbital Fock operator as... 

For a closed-shell system, the spin-up and spin-down Fock operators are equal and the spin-orbitals are obtained in pairs of equal shape and energy. Instead of dividing the set of orbitals into spin-up and spin-down orbitals, it is also possible to pursue a division into closed-shell and open-shell orbitals. This leads to the restricted open-shell HF (ROHF) method [23-25]. The formalism for this method is slightly more involved than the UHF formalism, but the general ideas are identical. The ROHF wavefunction is an eigenfunction of the total spin squared (5 ) operator, while the UHF wavefunction does not have this feature. The energy of the UHF wavefunction, on the other hand, is lower than that of the ROHF wavefunc-... 

The FCI wave function is often dominated by a single reference configuration, usually the Hartree-Fock state. It is then convenient to think of the FCI wave function as generated from this reference configuration by the application of a linear combination of spin-orbital excitation operators... 

The HF [31] equations = e.cj). possess solutions for the spin orbitals in T (the occupied spin orbitals) as well as for orbitals not occupied in F (the virtual spin orbitals) because the operator is Flennitian. Only the ( ). occupied in F appear in the Coulomb and exchange potentials of the Fock operator. 

As fonnulated above, the FIF equations yield orbitals that do not guarantee that F has proper spin symmetry. To illustrate, consider an open-shell system such as the lithium atom. If Isa, IsP, and 2sa spin orbitals are chosen to appear in F, the Fock operator will be... 

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

More abstractly the condition Tr (p+p ) = iV implies that the part of the Hilbert space defined by the projection operator p should be fully contained in the part defined by the projection operator p+. If we now vary p slightly so that this condition is no longer fulfilled, Eq. II.GO shows that the pure spin state previously described by the Slater determinant becomes mixed up with states of higher quantum numbers S = m+1,. . . . The idea of the electron pairing in doubly occupied orbitals is therefore essential in the Hartree-Fock scheme in order to secure that the Slater determinant really represents a pure spin state. This means, however, that, in the calculation of the best spin orbitals y>k(x), there is a new auxiliary condition of the form... 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

The terms on the right-hand side of eq. (11.41) denote the kinetic energy, the electron-nuclear potential energy, the Coulomb (J) and exchange (K) terms respectively. Together J and K describe an effective electron-electron interaction. The prime on the summation in the expression for K exchange term indicates summing only over pairs of electrons of the same spin. The Hartree-Fock equations (11.40) are solved iteratively since the Fock operator / itself depends on the orbitals iff,. 

The structure of the parametric UA for the 4-RDM satisfies the fourth-order fermion relation (the expectation value of the commutator of four annihilator and four creator operators [26]) for any value of the parameter which is a basic and necessary A-representability condition. Also, the 4-RDM constructed in this way is symmetric for any value of On the other hand, the other A-representability conditions will be affected by this value. Hence it seems reasonable to optimize this parameter in such a way that at least one of these conditions is satisfied. Alcoba s working hypothesis [48] was the determination of the parameter value by imposing the trace condition to the 4-RDM. In order to test this working hypothesis, he constructed the 4-RDM for two states of the BeHa molecule in its linear form Dqo/,. The calculations were carried out with a minimal basis set formed by 14 Hartree-Fock spin orbitals belonging to three different symmetries. Thus orbitals 1, 2, and 3 are cr orbitals 4 and 5 are cr and orbitals 6 and 7 are degenerate % orbitals. The two states considered are the ground state, where... 

The first set of equations govern the Cj amplitudes and are called the CI- secular equations. The second set determine the LCAO-MO coefficients of the spin-orbitals (f>j and are called the Fock equations. The Fock operator F is given in terms of the one- and two-electron operators in H itself as well as the so-called one- and two-electron density matrices yij and Tyj i which are defined below. These density matrices reflect the averaged occupancies of the various spin orbitals in the CSFs of VP. The resultant expression for F is ... 

Summation over repeated upper and lower indices is assumed = p f q) and — pq v sr) are the one- and two-electron integrals in a molecular spin-orbital basis p corresponding to the Fock operator (/) and the two-body part of the Hamiltonian (v). 

It is worth noting that the methods mentioned give rise to Hartree-Fock equations based on different Fock operators for the various orbitals of the same spin. This implies that off-diagonal Lagrange multipliers appear which cannot be eliminated by a suitable unitary transformation. Moreover, the complications arise when excited states above the first one are considered (see, e.g. [25] for further details). 

Functionals. The difference between the Fock operator, in wave function based calculations and the analogous operator in DFT calculations is that the Coulomb and exchange operators in T are replaced in DFT by a functional of the electron density. In principle, this functional should provide an exact formula for computing the Coulombic interactions between an electron in a KS orbital and all the other electrons in a molecule. To be exact, this functional must include corrections to the Coulombic repulsion energy, computed directly from the electron density, for exchange between electrons of the same spin and correlation between electrons of opposite spin. 

In the usual formalism of quantum mechanics, the first quantization formalism, observables are represented by operators and the wave functions are normal functions. In the method of second quantization, the wave functions are also expressed in terms of operators. The formalism starts with the introduction of an abstract vector space, the Fock space. The basis vectors of the Fock space are occupation number vectors, with each vector defined by a set of occupation numbers (0 or 1 for fermions). An occupation number vector represents a Slater determinant with each occupation number giving the occupation of given spin orbital. Creation and annihilation operators that respectively adds and removes electrons are then introduced. Representations of usual operators are expressed in terms of the very same operators. 

The occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

We now define an annihilation operator, a acting on the elements of the Fock space, such that it removes an electron from spin-orbital i to give a ket with N-l electrons occupied ... 

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