Second quantized form

These ideas can be applied to electrochemical reactions, treating the electrode as one of the reacting partners. There is, however, an important difference electrodes are electronic conductors and do not posses discrete electronic levels but electronic bands. In particular, metal electrodes, to which we restrict our subsequent treatment, have a wide band of states near the Fermi level. Thus, a model Hamiltonian for electron transfer must contains terms for an electronic level on the reactant, a band of states on the metal, and interaction terms. It can be conveniently written in second quantized form, as was first proposed by one of the authors [Schmickler, 1986] ... 

Here the operator af creates (and the operator a, removes) an electron at site i the nn denotes near-neighbors only, and /i,y = J drr/),/l(j)j denotes a Coulomb integral if i = j and a resonance integral otherwise. The second quantization form of this equation clearly requires a basis set. It is a model for the behavior of benzene - not a terribly accurate one, but one that helps us understand many things about its spectroscopy, its stability, its binding patterns, and other physical and chemical properties. 

In (4.28) and (4.30), we have achieved our aim of expressing the Hamiltonian in the appropriate second quantized form for acting on the state vectors in Fock space. 

The operators W, A, occurring above, should be taken in the second-quantization form, free of explicit dependence on particle number, and Tr means the trace in Fock space (see e.g. [10] for details). Problems of existence and functional differentiability of generalized functionals F [n] and r [n] are discussed in [28] the functional F [n] is denoted there as Fi,[n] or Ffrac[n] or FfraoM (depending on the scope of 3), similarly for F [n]. Note that DMs can be viewed as the coordinate representation of the density operators. 

The Dirac-Coulomb-Breit Hamiltonian H qb 1 rewritten in second-quantized form [6, 16] in terms of normal-ordered products of spinor creation and annihilation operators r+s and r+s+ut, ... 

An approach to constructing CSFs and matrix elements of the Hamiltonian that initially appears quite different from the symmetric group approach can be developed by considering the second-quantized form of the Hamiltonian. If we have an orthonormal... 

Many-electron wave functions in second-quantization form can conveniently be represented in an operator form. To this end, we shall introduce the vacuum state 0), i.e. the state in which there are no particles. We shall define it by... 

Equations (13.22) and (13.23) define the second-quantization form of an operator corresponding to a physical quantity, if its matrix elements are known in coordinate representations ((13.24) and (13.25)). Specifically, the operator of the total number of particles in a system will be... 

This expression is the second-quantization form of an arbitrary two-electron operator with tensorial structure of the kind (14.57). Examination of this formula enables us to work out expressions for operators that correspond to specific physical quantities. Many of these operators possess a complex tensorial structure, and their two-electron submatrix elements have a rather cumbersome form [14]. Therefore, by way of example, we shall consider here only the most important of the two-electron operators - the operator of the energy of electrostatic interaction of electrons (the last term in (1.15)). If we take into account the tensorial structure of that operator and put its submatrix element into (14.61), we arrive at... 

Next we shall represent (17.5) in the second-quantized form... 

Atomic spectral moments can be expressed in terms of the averages of the products of the relevant operators. Let Oi,C>2, --,Ok be the operators of interactions in definite shells or the operators of electronic transitions between definite shells in the second quantization form. The average of... 

The Ms and Mv quantum numbers of a given 2S+1Z/ term can be raised or lowered by using the well known shift operators JS and . The second-quantized form of these operators is as follows ... 

This hamiltonian has cylindrical symmetry and may be used to introduce trigonal or tetragonal anisotropy, depending on whether the principal z axis is oriented along a C3 or C4 symmetry axis. The second-quantized form of the intra-r29 part of this operator is given in Eq. 39. 

Both yDi and Jtsoc will connect these components to the 2P term. Using the second-quantized form of these operators, as specified in Eqs. 38 and 51, one... 

For applications the tunneling Hamiltonian (54) should be formulated in the second quantized form. We introduce creation and annihilation Schrodinger operators c fc, cRk, (hq, cRq. Using the usual rules we obtain... 

These moves allow us to write the electronic Hamiltonian in the second quantized form with respect to the basis of (spin-)orbitals ok (x) introduced above ... 

The PPP Hamiltonian can then be written in second quantized form as... 

Expanding the wave function in a linear combination of pure spin functions could yield the correct secular equations and thus correct eigenvalues. However, such spin-only wave functions could not be considered complete since complete wave functions must describe both the spatial and spin motions of electrons and must be antisymmetric under exchange of any two electrons. It would be better to rewrite the VB model (18) in the second quantization form as given in Eq. (20), in which its eigenstates can be taken as a linear combination of Slater determinants or neutral VB structures. Then... 

The one- and two-electron integrals appearing in the second-quantized form of the Hamiltonian carry all information about the specific features of the quantum system. The one-electron integrals are defined as... 

This general notation is deceptively simple. The bra is an excited determinant. There is an equation for each excited determinant, and each level of excitation leads to a different type of equation. Furthermore, the equations are all coupled, and they are non-linear in the amplitudes. However, they may be formulated in a quasilinear manner [27], and they have been solved for a wide range of CC schemes. The operator HN is the Hamiltonian written in second-quantized form minus the energy of the reference determinant, i.e. HN = H— < 0 /7 0 >. The subscript C restricts the operator product of HN and eT to connected terms. Once the CC equations have been solved, the CC correlation energy can be calculated from... 

The formulation of the relativistic CASPT2 method is almost the same as the nonrelativistic CASPT2 in the second quantized form. In this section, firstly we express the relativistic Hamiltonian in the second quantized form, and then, we give a summary of the CASPT2 method [11, 12],... 

It is more convenient to express the different contribntions in second quantized form. Thns, we have for the electrode and its interaction with the reactant ... 

The change of the proton potential on the hydrogen bond caused by the antiphase vibrations of the oxygens results in the shortening of the bond length. Hence the modes j = 2 and j = 4 are treated as having the coordinates of polarization vectors 112 and 114 approximated by their values at k = 0. Thus the interaction of the protons with the oxygen vibrations can be represented in the second quantization form... 

In the above equations, hpv are the usual one-electron integrals while [juv Ao] and [juA vo] are the standard bare and antisymmetrized two-electron integrals, respectively. To derive these formulae, one has merely to substitute the second quantized form of the total Hamiltonian and apply the above rules for the density matrix elements. The analogy of Eq. (27) to the corresponding HF formula is obvious. 

---