Electronic Density Matrix

The density matrix corresponding to an orbital product is defined as 

For 1 = V, we obtain p(l) = Yo(l l). equal to the electronic density. In Equation 1.96, 1 and Y are different in order to be able to operate on only the right function, as is required when an expectation value is calculated. 

Generally, y(1 10 may be calculated from an arbitrary wave function using the definition (Lowdin 1955)  

To be able to calculate two-electron expectation values, knowledge of Y(1 10 is insufficient. To obtain the same simplification as for the one-electron expectation values, the two-matrix, a function of 1, T, 2, and 2, might be introduced, but this is not considered here. The two-matrix contains all necessary information, but unfortunately the two-matrix cannot easily be obtained without first calculating the wave function (the N-representability problem). 

We note that the following relation holds in the case when T is a Slater determinant  

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We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

When working with atomic orbitals, it is usual to write the electron density in terms of a certain matrix called (not surprisingly) the electron density matrix. For the simple dihydrogen VB wavefunction, we have... 

This shows that, when we have found the correct electron density matrix and correctly calculated the Hartree-Fock Hamiltonian matrix from it, the two matrices will satisfy the condition given. (When two matrices A and B are such that AB = BA, we say that they commute.) This doesn t help us to actually find the electron density, but it gives us a condition for the minimum. 

The a and yS electrons are considered separately, and there is an electron density matrix for each set. These add to give the electron density, whilst the difference is the spin density. 

The calculation usually proceeds along the traditional lines of HF-LCAO theory. We make an initial estimate of the electron density matrix P, calculate a revised h and iterate until the electron density and the HP matrix are self-consistent. 

The concept of natural orbitals may be used for distributing electrons into atomic and molecular orbitals, and thereby for deriving atomic charges and molecular bonds. The idea in the Natural Atomic Orbital (NAO) and Natural Bond Orbital (NBO) analysis developed by F. Weinholt and co-workers " is to use the one-electron density matrix for defining the shape of the atomic orbitals in the molecular environment, and derive molecular bonds from electron density between atoms. 

The treatment presented so far is quite general and formally exact. It combines the eikonal representation for nuclear motions and the time-dependent density matrix in an approach which could be named as the Eik/TDDM approach. The following section reviews how the formalism can be implemented in the eikonal approximation of short wavelengths for the nuclear motions, and for specific choices of electronic states leading to the TDHF equations for the one-electron density matrix, and to extensions of TDHF. 

This in turn leads to the initial one-electron density matrix by contraction over electron variables,(48)... 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

This method relies on the use of an auxiliary gaussian basis set to fit the molecular electron density obtained from an ab initio one-electron density matrix ... 

A measure for the electronic coherence of the wave function are the nondiagonal elements of the electronic density matrix, which, for example, in the diabatic representation are given by (k k )... 

Figure 15. Average number of random walkers generated for a single iteration as obtained for Model IVa [205], The full and short dashed lines correspond to the upper and lower electronic populations, respectively, while the long dashed line corresponds to the sum of the coherences of the electronic density matrix.

Integrating the A -electron density matrix over coordinates 3 to N generates the two-electron density matrix (2-RDM) ... 

Both the energy as well as the one- and two-electron properties of an atom or molecule can be computed from a knowledge of the 2-RDM. To perform a variational optimization of the ground-state energy, we must constrain the 2-RDM to derive from integrating an A -electron density matrix. These necessary yet sufficient constraints are known as A -representability conditions. 

The 2-RDM/or the radical may be computed from the N + l)-electron density matrix for the dissociated molecule by integrating over the spatial orbital and spin associated with the hydrogen atom and then integrating over N — 2 electrons. Because the radical in the dissociated molecule can exist in a doublet state with its unpaired electron either up or down, that is, M = the 2-RDM for the radical is an arbitrary convex combination... 

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... 

J. M. Herbert, Reconstructive Approaches to One- and Two-Electron Density Matrix Theory, Ph.D. thesis, University of Wisconsin, Madison, WI, 2003. 

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... 

In a famous after-dinner address at a 1959 conference in Boulder, Colorado, Charles Coulson [1] discussed both the promises and the challenges of using the two-electron reduced density matrix (2-RDM) rather than the many-electron wavefunction as the primary variable in quantum computations of atomic and molecular systems. Integrating the A-electron density matrix,... 

This procedure is simple and robust and applies equally to a many-group system. Moreover, the group functions may be of arbitrary form it is necessary only that the 1-electron density matrix can be calculated for each one. In the present context, for example, a core function of Hartree-Fock form may be used along with a valence function of VB form, thus allowing for correct dissociation of the system when its constituent groups are removed to infinity. 

Clinton WL, Frishberg C, Massa LJ, Oldfield PA (1973) Methods for obtaining an electron-density matrix from X-ray diffraction data. Int J Quantum Chem Symp 7 505-514... 

The two-electron density matrix elements are given in similar fashion ... 

This is the well-known full-CI scheme. The two-electron density matrix is defined by the formula... 

Eventually, quantities D can be considered as the Ath element of vector 5, so we can write the element of two-electron density matrix 0 as a scalar product of two vectors ... 

The expressions Eqs. (2), (4) are completely general. To address the aspects important for the TMCs modelling, i.e. the energies of the corresponding electronic states, we notice that the statement that the motion of electrons is correlated can be given an exact sense only with use of the two-electron density matrix Eq. (4). Generally, it looks like [35] (with subscripts and variables notations w omitted for brevity) ... 

The attempts to construct an acceptable parameterization for TMCs are almost exclusively undertaken within the framework of the HFR MO LCAO paradigm. It is easy to understand that the nature of failures which accompany this direction of research as long as it exists lays precisely in the inadequate treatment of the cumulant of the two-electron density matrix by the HFR MO LCAO. 

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