Orbital excitation operators

In the previous sections, the occupation number vectors were specified in terms of the occupation of a set of spin orbitals, and the operators were defined by integrals over spin orbitals multiplied with spin orbital excitation operators. The spin orbitals depend on a continuous spatial coordinate, r, and a discrete spin coordinate ms. The spin coordinate takes two values, so the complete spin basis is spanned by two functions a(ms), a = a, p defined as... 

So far we have based the formalism on an N-electron basis built from Slater determinants. However, as shown above, both the Hamiltonian and the orbital rotations can be described in terms of the orbital excitation operators Ey. All... 

In multiconfiguration response theory, the space of excited states, the orthogonal complement of the reference state, is described in terms of operators acting on the reference state orbital excitation operators... 

The so-called cluster operator T is expressed in terms of spin-orbital excitation operators of the form Ti = p a, T2 = p q ba, T = p q r cba, etc. with relating to the excitation of k electrons from occupied spin-orbitals (a, b, c, etc.) to virtual spin-orbitals (p,q,r, etc.). Prior to forming any EA EOM, the neutral molecule CC equations need to be solved for the amplimdes t that multiply the r operators to form the CC T operator. Eor completeness, let us briefly review how the conventional CC wave function evaluation is carried out. 

By contrast, CASPT2 uses orbital excitation operators applied to the root wavefunction to express the perturbation function. Apart from the root function itself, this generates the singly excited wavefunctions the doubly excited wavefunctions and so on. These... 

The incorporation of spin in second quantization leads to operators with different spin synunetry properties as demonstrated in Section 2.2. Thus, spin-free interactions are represented by operatOTs that are totally symmetric in spin space and thus expressed in terms of orbital excitation operators that affect alpha and beta electrons equally, whereas pure spin interactions are represented by excitation operators that affect alpha and beta electrons differently. For the efficient and transparent manipulation of these operators, we shall apply the standard machinery of group theory. More specifically, we shall adopt the theory of tensor operators for angular momentum in quantum mechanics and develop a useful set of tools for the construction and classification of states and operators with definite spin symmetry properties. 

Except for a scaling factor, the singlet excitation operator (2.3.21) is identical to the orbital excitation operator in (2.2.7) ... 

In Section 1.7, we discussed density matrices in the spin-orbital basis. In the orbital basis, the density matrices are conveniently defined in terms of the spin tensor excitation operators of Section 2.3.4 rather than in terms of the spin-orbital excitation operators of Section 1.3.3. We shall in this section consider the properties of density matrices in the orbital basis, relating their properties to the elementary spin-orbital densities. The exposition in this section follows closely that of Section 1.7. 

As discussed in Section 4.4, the conservation of spin and spatial symmetries in the optimized wave function is then guaranteed only if the symmetry restrictions are explicitly imposed on the parametrization. In other words, the spin and spatial symmetries are conserved only if k contains only those spin-orbital excitation operators that transform as the totally symmetric representation of the Hamiltonian H. If nontotally symmetric rotations are allowed, then 0> will not have the symmetry of 0) and the symmetries of 0) and 0) will be different. 

When a perturbation V is applied to the system, the Hamiltonian becomes H + V. The allowed variations then become those that transform according to the totally symmetric representation of H + V rather than of H. For example, if we consider the ground state of the oxygen molecule and if V corresponds to an electric field perpendicular to the intemuclear axis, then the allowed spin-orbital excitations are represented by spin-conserving orbital excitation operators that are totally symmetric in the C2v point group. In other words, the allowed variations are described by operators Ep where the direct product of the irreducible representations of the orbitals (t>p and (p is totally symmetric in C2v... 

Treating the nondiagonal orbital excitation operators in the same way, we arrive at the following expression for k ... 

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... 

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

In our formalism [5-9] excitation operators play a central role. Let an orthonormal basis p of spin orbitals be given. This basis has usually a finite dimension d, but it should be chosen such that in the limit —> cxd it becomes complete (in the so-called first Sobolev space [10]). We start from creation and annihilation operators for the ij/p in the usual way, but we use a tensor notation, in which subscripts refer to annihilation and superscripts to creation ... 

Most Hamiltonians of physical interest are spin-free. Then the matrix elements in Eq. (9) depend only on the space part of the spin orbitals and vanish for different spin by integration over the spin part. Then it is recommended to eliminate the spin and to deal with spin-free operators only. We start with a basis of spin-free orbitals cpp, from which we construct the spin orbitals excitation operators carry orbital labels (capital letters) and spin labels... 

Greek letters). We define spin-free excitation operators carrying only orbital labels, by summation over spin... 

As in the previous section we consider a single Slater determinant reference function with the spin orbitals i/, occupied. However, we express our excitation operators in a completely arbitrary basis of spin orbitals i/, which is no longer the direct sum of occupied and unoccupied spin orbitals. Then the following replacements must be made [3] ... 

I being the identity matrix. Introducing an explicit notation for the excitation operators, which specifies the hole and particle (spin) orbitals as, respectively, the subscripts and superscripts,... 

These operator relations allow manipulating the operators independently of the function they are operating on. In general we will work with products of the operators. These can then often be simplified by the use of (3 5) or relations derived from them. Important operator products are those that preserve the number of particles. They always contain equally many annihilation and creation operators. A basic operator of this kind is the single excitation operator, which excites an electron from orbital i to orbital j ... 

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