Hamiltonian, second-quantized operators

The next step is that we find inverse transformations to (25-28) and substitute these inverse transformations into eq. (22) and then applying Wick theorem, we requantize the whole Hamiltonian (16) in a new fermions and bosons [14]. This leads to new V-E Hamiltonian (we omit sign on the second quantized operators)... 

In previous part we developed canonical transformation (through normal coordinates) by which we were able to pass from crude adiabatic to adiabatic Hamiltonian. We started with crude adiabatic molecular Hamiltonian on which we applied canonical transformation on second quantized operators... 

The reduced matrices and V represent a partitioning of the Hamiltonian into one- and two-electron parts. Rearranging the second-quantized operators and using the definition of the 2- and 3-RDMs,... 

The Hamiltonian matrix in Equation (15) is obtained from appropriate products of representations of second-quantized operators that act within the left block, right block, or partition orbital. For example, in the case of where... 

In this expression, hp = pW q) represeiits a matrix element of the one-electron component of the Hamiltonian, h, while (pqWrs) s ( lcontains general annihilation and creation operators (e.g., or ) that may act on orbitals in either occupied or virtual subspaces. The cluster operators, T , on the other hand, contain operators that are restricted to act in only one of these spaces (e.g., al, which may act only on the virtual orbitals). As pointed out earlier, the cluster operators therefore commute with one another, but not with the Hamiltonian, f . For example, consider the commutator of the pair of general second-quantized operators from the one-electron component of the Hamiltonian in Eq. [53] with the single-excitation pair found in the cluster operator, Tj ... 

The Kronecker delta functions, 5 and 6,p, resulting from Eq. [21], cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of H and T serves to eliminate one of the electronic Hamiltonian s general-index annihilation or creation operators in favor of a simple delta function. Since f contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from f will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher order terms will contain commutators of only the cluster operators, T, and are therefore zero. Hence, Eq. [52] truncates itself naturally after the first five terms on the right-hand side. ° This convenient property results entirely from the two-electron property of the Hamiltonian and from the fact that the cluster opera-... 

In a non-relativistic theory we would now continue by adding a second quantized operator for two-body interactions. In the relativistic case we need to step back and first consider the interpretation of the eigenvalues of the Hamiltonian. Dirac stated that positrons could be considered as holes in an infinite sea of electrons . In this interpretation the reference state for a system with neither positrons nor electrons is the state in which all negative energy levels are filled with electrons. This vacuum state... 

To be able to apply this quasiparticle framework we should derive the quasiparticle representation of the Hamiltonian. The first step along this line is to substitute the ordinary second-quantized operators by quasiparticles using the inverse of Eqs. (5) and (6) as... 

Within second quantization [41] the Hamiltonian operator may be expressed as... 

By Eq. (6) the sum on the right-hand side of the above equation is equal to the energy E, and from Eq. (2) we realize that the sums on the left-hand side are just Hamiltonian operators in the second-quantized notation. Hence, when the 2-RDM corresponds to an A -particle wavefunction i//, Eq. (12) implies Eq. (13), and the proof of Nakatsuji s theorem is accomplished. Because the Hamiltonian is dehned in second quantization, the proof of Nakatsuji s theorem is also valid when the one-particle basis set is incomplete. Recall that the SE with a second-quantized Hamiltonian corresponds to a Hamiltonian eigenvalue equation with the given one-particle basis. Unlike the SE, the CSE only requires the 2- and 4-RDMs in the given one-particle basis rather than the full A -particle wavefunction. While Nakatsuji s theorem holds for the 2,4-CSE, it is not valid for the 1,3-CSE. This foreshadows the advantage of reconstructing from the 2-RDM instead of the 1-RDM, which we will discuss in the context of Rosina s theorem. 

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, ... 

Here h are the one-electron integrals including the electron kinetic energy and the electron-nuclear attraction terms, and gjjkl are the two-electron repulsion integrals defmed by (3 19). The summations in (3 24) are over the molecular orbital basis, and the definition is, of course, only valid as long as we work in this basis. Notice that the number of electrons does not appear in the defmition of the Hamiltonian. All such information is found in the Slater determinant basis. This is true for all operators in the second quantization formalism. 

In order to be able to write out all the terms of the direct Cl equations explicitly, the Hamiltonian operator is needed in a form where the integrals appear. This is done using the language of second quantization, which has been reviewed in the mathematical lectures. Since, in the MR-CI method, we will generally work with spin-adapted configurations a particularly useful form of the Hamiltonian is obtained in terms of the generators of the unitary group. The Hamiltonian in terms of these operators is written,... 

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... 

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. 

We now introduce creation and annihilation operators ajj and an which create/annihilate e-h pairs at a given combination of sites n = (n, n1), i.e., 41°) = 14 = nen h), where 0) is the ground state. Using these operators, a generic monoexcitation configuration interaction Hamiltonian can be formulated as follows in second quantization notation,... 

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... 

One of the most important concepts of quantum chemistry is the Slater determinant. Most quantum chemical treatments are made just over Slater determinants. Nevertheless, in many problems the formulation over Slater determinants is not very convenient and the derivation of final expressions is very complicated. The advantage of second quantization lies in the fact that this technique permits us to arrive at the same expressions in a considerably simpler way. In second quantization a Slater determinant is represented by a product of creation and annihilation operators. As will be shown below, the Hamiltonian can also be expressed by creation and annihilation operators and thus the eigenvalue problem is reduced to the manipulation of creation and annihilation operators. This manipulation can be done diagrammatically (according to certain rules which will be specified later) and from the diagrams formed one can write down the final mathematical expression. In the traditional way a Slater determinant I ) is specified by one-electron functions as follows ... 

Wick s theorem (35) which gives us the prescription for treating a product of operators may of course be applied to the Hamiltonian, expressed in the second quantization formalism (29). This leads to the Hamiltonian in a form which is of primary importance in perturbation treatments. This form of the Hamiltonian which is called the normal product form is ... 

H is the Hamiltonian operator expressed in the language of second quantization,... 

This representation among others removes one more inconsistency in quantum chemistry one generally deals with the systems of constant composition i.e. of the fixed number of electrons. The expression eq. (1.178) allows one to express the matrix elements of an electronic Hamiltonian without the necessity to go in a subspace with number of electrons different from the considered number N which is implied by the second quantization formalism of the Fermi creation and annihilation operators and on the other hand allows to keep the general form independent explicitly neither on the above number of electrons nor on the total spin which are both condensed in the matrix form of the generators E specific for the Young pattern T for which they are calculated. 

Closely inspecting the operator terms entering the electronic Hamiltonian eq. (1.27) one can easily see that they are sums of equivalent contributions dependent on coordinates of one or two electrons only. Analogously in the second quantization formalism only the products of two and four Fermi operators appear in the Hamiltonian. Inserting the trial. Y-electron wave function of the (ground) state into the expression for the electronic energy yields its expectation value in terms of the expectation values of the one- and two-electron operators ... 

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