Density matrices normalization

The term containing Dirac s delta ZaS(r — Ra) represents the contribution from the positive point charge Za at the position Ra of the nucleus a, and -pA(r) is the electronic charge distribution, given by the diagonal element of the first-order density matrix normalized to the number of electrons in the monomer A. 

The orbital occupation numbers n, (eigenvalues of the density matrix) will be between 0 and 1, corresponding to the number of electrons in the orbital. Note that the representation of the exact density normally will require an infinite number of natural orbitals. The first N occupation numbers N being the total number of electrons in the system) will noraially be close to 1, and tire remaining close to 0. 

Population analysis with semi-empirical methods requires a special comment. These methods normally employ the ZDO approximation, i.e. the overlap S is a unit matrix. The population analysis can therefore be performed directly on the density matrix. In some cases, however, a Mulliken population analysis is performed with DS, which requires an explicit calculation of the S matrix. 

The diagonal element of the density matrix, W(n) — a nan is the probability that a system chosen at random from the ensemble occurs in the state characterized by n, and implies the normalization... 

Here N is the normalization constant and the coefficients Ai,A2 and B will be determined by Schrodinger equation. We are interested in the reduced density matrix for the long wavelength mode , which is obtained by integrating the density matrix To(i, 2, t) over the short wavelength mode 2 = reduced density matrix for the long wavelength mode can be written in the form... 

The Af-th order density matrix (DM) generated from a normalized wave function of a Af-electron system is defined as... 

Let the eigenvalue w be fixed and assume that fit is nondegenerate and unit-normalized. The restriction to nondegenerate eigenstates will be relaxed in Section V, but for now we consider only pure-state density matrices. The A -electron density matrix for the pure state fit is... 

Formulating conditions for the energy to be stationary with respect to variations of the wavefunction P in this generalized normal ordering, one is led to the irreducible Brillouin conditions and irreducible contracted Schrodinger equations, which are conditions on the one-particle density matrix and the fe-particle cumulants k, and which differ from their traditional counterparts (even after reconstruction [4]) in being strictly separable (size consistent) and describable in terms of connected diagrams only. 

For the formulation of the generalized Wick theorem corresponding to the generalized normal ordering, we need the matrix element rf, Eq. (65), of the one-hole density matrix and the cumulants kjc, Eqs. (39)-(47), of the fc-particle density matrices. 

The n-particle density matrix of an w-particle state is pure-state n-representable if—for unit trace—it is idempotent. Since we normalize y as... 

Note that this ensemble includes contributions from different choices of ( ii I I Note also that the N-electron density matrix has been defined so that it is normalized to one.) This convex set of N-electron density matrices can be reduced to a convex set of /f-electron reduced density matrices using the definition... 

Let f, P and f, P be (2/ + 1) x 1 matrices representing the density-function normalized spherical harmonics and their population parameters, before and after rotation, respectively. Then, by using Eq. (D.10), we construct a (21 + 1) x (21 + 1) matrix M such that... 

We recognize this as the normal Fock matrix in closed shell Hartree-Fock theory, but only for those matrix elements where the second index, i, corresponds to an occupied orbital. The first index is arbitrary. The condition for an optimum wave function is then Fpi = Fip. This is trivially fulfilled when also p corresponds to an occupied orbital, since the wave function is invariant towards rotations among these orbitals. On the other hand, when p is a virtual orbital we have Fip = 0 (all density matrix elements in (4 42) are then zero). 

The solution of this equation would give the time-evolution of all system as well as bath degrees of freedom. This is often unnecessary and leads to a huge number of equations. Therefore one normally introduces the reduced density matrix p(t) of the system degrees of freedom only. 

Equation 5.78 is normally modified by subsuming the c s into Ptu, the elements of the density matrix P ... 

It is convenient to convert to the coordinate representation of the density matrix of the vibration system to take into account the possible change of the set of normal coordinates. The Hamiltonian of the harmonic vibration system in the initial and final states may be expressed in the form ... 

In Eq. (10), E nt s(u) and Es(in) are the s=x,y,z components of the internal electric field and the field in the dielectric, respectively, and p u is the Boltzmann density matrix for the set of initial states m. The parameter tmn is a measure of the line-width. While small molecules, N<pure solid show well-defined lattice-vibrational spectra, arising from intermolecular vibrations in the crystal, overlap among the vastly larger number of normal modes for large, polymeric systems, produces broad bands, even in the crystalline state. When the polymeric molecule experiences the molecular interactions operative in aqueous solution, a second feature further broadens the vibrational bands, since the line-width parameters, xmn, Eq. (10), reflect the increased molecular collisional effects in solution, as compared to those in the solid. These general considerations are borne out by experiment. The low-frequency Raman spectrum of the amino acid cystine (94) shows a line at 8.7 cm- -, in the crystalline solid, with a half-width of several cm-- -. In contrast, a careful study of the low frequency Raman spectra of lysozyme (92) shows a broad band (half-width 10 cm- -) at 25 cm- -,... 

Other forms of normalization, as well as forms denoting irreducible tensor operators may be found in [304] and in Appendix D. With the aid of the orthogonality relation one may easily express the quantum mechanical polarization moments fq and Pq through the elements of the density matrix /mm and... 

Since the density matrix is Hermitian, we obtain the property of polarization moments which is analogous to the classical relation (2.15) fq = (—1 ) (f-q) and tp = (—l) 3( g). The adopted normalization of the tensor operators (5.19) yields the most lucid physical meaning of quantum mechanical polarization moments fq and p% which coincides, with accuracy up to a normalizing coefficient that is equal for polarization moments of all ranks, with the physical meaning of classical polarization moments pq, as discussed in Chapter 2. For a comparison between classical and quantum mechanical polarization moments of the lower ranks see Table 5.1. 

In the present book we have used the cogredient expansion form (2.14), where, as distinct from the standard form, an additional normalizing factor has been introduced, namely (—l)< v/(2K + l)/4n. Our expansion of the classical probability density p(0, differs from the standard one in exactly the same way as the expansion of the quantum mechanical density matrix p over 2Tq differs from the expansion over lTg. In Section 5.3 we present a comparison between the physical meaning of the classical polarization moments pg, as used in the present book, and the quantum mechanical polarization moments fg, as determined by the cogredient method using normalization (D.ll). 

The strong role of collisions in decohering a system is readily seen by considering fie density matrix pj of a system that has reached thermal equilibrium at tempera-rs T through collisional relaxation, that is, pf = Qexp(—Hs./kgT). Here Hs is the system Hamiltonian, Q is a normalization factor, and kB is the Boltzmann constant, fmsiderable insight is obtained if we cast the density matrix in the energy repre-... 

The resulting density matrix <73 is just the sum of the enhanced (by a factor of 4) and antiphase 13C single quantum coherence and the normal in-phase 13C SQC. 

The N particle (and its p-reduced companions) representable density matrix T(p) can be defined as follows in the Lowdin normalization (x, is a combined space-spin coordinate)... 

The transformation of the density matrix by a 90° pulse and its subsequent evolution as the magnetization precesses freely depend on whether the pulse is applied only to one nucleus or to both I and S. We treat first the situation that would normally occur in a heteronuclear system, where a 90x pulse is applied only to the I spins—a 90pulse. This treatment is, of course, also applicable to a homonu-clear system subjected to a selective rf pulse. 

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