Electronic states expectation values

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... 

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... 

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... 

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... 

There exists a hierarchy of Green s functions (GF). These functions are defined as the //-electron ground-state expectation value of a time-ordered product of annihilation and creation operators. The simplest and probably most relevant member of the hierarchy is the one-particle GF... 

QMC teclmiques provide highly accurate calculations of many-electron systems. In variational QMC (VMC) [112, 113 and 114], the total energy of the many-electron system is calculated as the expectation value of the Hamiltonian. Parameters in a trial wavefiinction are optimized so as to find the lowest-energy state (modem... 

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenlieimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a fiinction of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefiinction parameters are most often detennined by tire variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function ... 

The SSH model (Eq. (3.2)) is, essentially, the model used by Peierls for his discussion of the electron-lattice instability [33]. Its ground state is characterized by a non-zero expectation value of the operator. 

The radial distribution ftinctions may be used to calculate expectation values of functions of the radial variable r. For example, the average distance of the electron from the nucleus for the Is state is given by... 

Expression (6.71) for the expectation value of may be used to calculate the average potential energy of the electron in the state nlm). The potential energy V r) is given by equation (6.13). Its expectation value is... 

Show explicitly for a hydrogen atom in the Is state that the total energy is equal to one-half the expectation value of the potential energy of interaction between the electron and the nucleus. This result is an example of the quantum-mechanical virial theorem. 

In words, we search over all allowed, antisymmetric N-electron wave functions and the one that yields the lowest expectation value of the Hamilton operator (i. e. the energy) is the ground state wave function. 

One can expect that the electron density corresponding to the electronic state of lowest energy is roughly constant in the interior of the metal and decreases to zero outside the metal. This means that the potential seen by an electron, due to the ion cores and the other electrons, is roughly constant inside the metal, with a value significantly lower than the potential outside. The simplest model for electrons in a metal, the Sommerfeld38 model, takes this potential as -V0 inside and 0 outside. One is then led to consider the one-dimensional Schrodinger equation... 

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