Ground-State Expectation Values

In Section 10.1 we will illustrate this for ground-state expectation values such as Eq. (4.25) and many others and in Section 10.2 for sum-over-states expressions such as Eq. (4.74) and many others. In the rest of the chapter we will discuss methods in which approximations are made to the exact matrix representation of the linear response function or polarization propagator given in Eq. (3.159). This equation is exact as long as a complete set of excitation and de-excitation operators hn is used and the reference state is an eigenfunction of the imperturbed Hamiltonian. 

Approximate polarization propagator methods are thus obtained by truncating the set of operators and by using an approximate reference state Mpller-Plesset 

All approximate ab initio methods presented in Chapter 9 are based on Slater determinants built with molecular orbitals. In this section we will therefore derive expressions for the expectation value ( o O o) of a general one-electron but spin-free operator O = o(0 an approximate closed-shell wavefunction o) in terms of the 

Starting from the expression for the ground-state expectation value, Eq. (3.46), as integral over the reduced one-electron density matrix P(r, r ) and using the expansion of it in molecular orbitals, Eq. (9.109), we obtain an expression for the expectation value as a contraction of the density matrix in the molecular orbital basis and molecular property integrals 

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These moments are related to many physical properties. The Thomas-Kulm-Reiche sum rule says that. S (0) equals the number of electrons in the molecule. Other sum rules [36] relate S(2),, S (1) and. S (-l) to ground state expectation values. The mean static dipole polarizability is md = e-S(-2)/m,.J Q Cauchy expansion... 

An important property of the dimerized Peierls stale is the existence of gaps in the spectra of spin and charge excitations. For free electrons (//ci-ci=0) both gaps are equal, while in the presence of Coulomb repulsion the spin gap is smaller than the charge gap [23, 24]. In what follows, we will assume the temperature to be much smaller than these two gaps, so that we can neglect electronic excitations and replace Hcl [ A (.v)] by its ground state expectation value. 

From the closure relation Z j j ) (j = 1 -1 g ) < g I, the sum over the product of transition matrix elements involving p,(r) and p (r )separates into two terms, one containing the ground-state expectation value of p (r) p (r ) and the other containing the product of the expectation values of p (r) and p (r ), both in the ground state. These terms can be further separated into those containing self interactions vs. those containing interactions between distinct electrons. Then... 

Consequently, 0) may be understood as a functional [/ ]> of the ground state four current. As an immediate result, y/ also determines all ground state expectation values, in particular the ground state energy E = [/ ]. 

Theorem D — The ground-state expectation value of s) s is positive or negative as i and j are in the same or different subsets, and... 

There are some further theorems concerning the signs of various ground-state expectation values (still for bipartite graphs). First [64] for expectations over an exchange operator for an arbitrary pair of sites... 

In the limit T —> 0, the free energy of the system, F = —ksTlnZ, becomes the ground-state energy and the various thermal averages become ground-state expectation values. 

Let us now pick an arbitrary density out of the set A of densities of nondegenerate ground states. The Hohenberg-Kohn theorem then tells us that there is a unique external potential v (to within a constant) and a unique ground state wavefunction I W[ri]) (to within a phase factor) corresponding to this density. This also means that the ground state expectation value of any observable, represented by an operator O. can be regarded as a density functional... 

Any ground state expectation value corresponding to an observable O is a functional of the density according to... 

We now consider the effect of small changes in the external field on the expectation values of physical observables. This is exactly what is studied in most experimental situations where one switches on and off an external field and studies how the system reacts to this. We will here study a more specific case in which we look at static changes in the external potential and their accompanying changes in the ground state expectation values. By investigating this problem we will learn how to take... 

Both difficulties with (3) are overcome by taking the ground-state expectation value of (3) to produce... 

Sauer " introduced an SOS expression for the DSO term. By defining a suitable operator and making use of a resolution of the identity in the state space in terms of eigenstates of the unperturbed Hamiltonian, the ground-state expectation value yielding the DSO term is replaced by an RSPT-like expression. A consistent correlated calculation of all J terms is thus possible. However, the DSO contribution may be calculated with a correlated ground-state wavefunction and therefore this method has not so far been used in practical applications. 

In the point-charge approximation the dipole operator commutes with any Hamiltonian term containing electron number operators or their products. Therefore only the electron transfer term ht contributes to the sum rule with its ground state expectation value, thus... 

Here indicates an infinitesimal positive time-shift of t, i.e. = lime o(t + e ). The KS and the full many-body Green s function are defined by the ground-state expectation values of the time-ordered product of the corresponding field operators, and... 

The second-order energy correction thus consists of two terms a ground-state expectation value over the second-order Hamiltonian and a so-called sum-over-states term, which involves a summation over all excited states of the system and transition moments between the ground state and these excited states with the first-order Hamiltonian Finally, we can insert the expressions for the first- and second-order perturbation Hamiltonians, Eq. (2.108),... 

Before we continue in the derivation of a matrix representation of the polarization propagator, we want to mention that by taking the zero-frequency limit of the equation of motion in the frequency domain, we obtain the following relation between a polarization propagator and a ground-state expectation value... 

The derivation of the induced contribution, on the other hand, is very similar to the derivation for the magnetizability. We could start from the definition of the rotational g tensor as first derivative of the rotational magnetic moment, Eq. (6.8), which would then be the induced contribution to it, and use the response theory formalism of Section 3.11. Using Eq. (3.116) we could express the derivatives of the induced rotational magnetic moment in terms of a polarization propagator and ground-state expectation value. Here we will, however, make use of the definition as second... 

The induced contribution consists therefore of a paramagnetic or sum-over-states contribution and a diamagnetic or ground-state expectation value term. Combining these with the contribution from the rigid charges, Eq. (6.15), yields... 

We are going to rewrite the three linear response functions now as ground-state expectation values similar to the derivations in Section 5.9. However, here we wiU not proceed via the sum-over-states expressions for the response function, but want to illustrate an alternative approach via the equation of motion of the polarization propagator for zero frequencies, Eq. (3.141). Recalhng that O p is the canonical conjugate... 

The electronic ground-state expectation values in the numerator of the vibrational polarizability are components of the permanent electric dipole moment, Elq. (4.40), and we can therefore write the vibrational polarizability more compactly as... 

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