First-order electron density

One of the most widely used definition of Molecular Similarity was origin-ally introduced by one of us [la,j] and it is defined in a quantum mechanical framework from the first order electron density functions Dt,Dj of the two molecules being compared using some form of MQSM as described in Eq. (14). 

In Eq. (10) j>a( i) represents the electron charge distribution, i.e. the diagonal element of the first-order electron density matrix which, in the SCF approximation c>, is given by... 

If a spinless -th order density function is desired for an N-electron molecule, integration should be performed over (N- ) spatial coordinate sets and all N spin variables. The first-order electron density in a point r is then ... 

Clearly, the electron density will depend on the method that obtains it. Nonetheless, even the simple Hartree-Fock (HF) electron density performs well to obtain first-order electron densities. The electron density in this MO LCAO framework is given by... 

As discussed earlier, we cannot only derive first-order electron densities, but also we can extend them to higher order electron densities. We have used the second-order electron density p(ri,r2) in lieu of the first-order electron density on several occasions in molecular quantum similarity, because the second-order electron density is in fact the lowest order density where electron correlation becomes apparent. It has been used extensively by Ponec et al. "- in the study of similarity in pericyclic reactions where the second-order electron density offers important advantages over the first-order electron density. In another contribution, Ponec et al. went to the third-order electron density. Again, most of the discussion relating to molecular quantum similarity indices and molecular alignment is also applicable to higher order electron densities, replacing where necessary the first-order electron density by, for example, the second-order electron density. 

The local aspect of the property, and its scalar characteristics, are both important in the process of reduction of the information, which is a basic point in the elaboration of interpretations of quantum systems and properties. The electronic components of i.e. is the diagonal terhi of the first order electronic density matrix, corresponding thus to a severe reduction of the information present in the original wavefunction. 

In Section 3.5 also the perturbed and in particular the first-order electron density Pa (r) and first-order reduced one-electron density matrix Pa r, P) were introduced. For these we can define corresponding first-order density matrices Da pq and... 

The electron density is the diagonal element of the number density matrix N(r,r ), i.e the first order redueed density matrix after integration over the spin coordinates, ... 

If electronic spin is not a focus of attention, then the spin-traced versions of these density matrices can be used. The r-space, spin-traced, first-order, reduced-density matrix is... 

The concept of the molecular orbital is, however, not restricted to the Hartree-Fock model. Sets of orbitals can also be constructed for more complex wave functions, which include correlation effects. They can be used to obtain insight into the detailed features of the electron structure. One choice of orbitals are the natural orbitals, which are obtained by diagonalizing the spinless first-order reduced density matrix. The occupation numbers (T ) of the natural orbitals are not restricted to 2, 1, or 0. Instead they fulfill the condition ... 

Davidson suggested that the wavefunction be projected onto a set of orbitals that have intuitive significance. These orbitals are a minimum set of atomic orbitals that provide the best least-squares fit of the first-order reduced-density matrix. Roby expanded on this idea by projecting onto the wavefunction of the isolated atom. One then uses the general Mulliken idea of counting the number of electrons in each of these projected orbitals that reside on a given atom to obtain the gross atomic population. 

If we are interested only in properties that can be expressed in terms of one-electron operators, then it is sufficient to work with the first-order reduced density matrix rather than the A-electron wave function [23-27]. 

Natural resonance theory (NRT) is an optimal ab initio realization of the resonance weighting concepts expressed in Equation 7.10 and Equation 7.11. The necessary and sufficient condition that Equation 7.10 be satisfied for all possible density-related (one-electron) properties p is that the first-order reduced density operator be expressible as such a weighted average of localized density operators t... 

In the ZDO approximation, the LHP method is well-adapted for computer programming. In fact, one can use any program for closed-shell molecules by making two small modifications firstly, the electron-density and bond-order matrix is defined for radicals as follows... 

This is generally true and, for example, is the reason why in the CPHF equations for magnetic perturbations there is only an exchange contribution and no Coulomb contribution due to the perturbed orbitals. The first-order FIF density vanishes, while the first-order HF density matrix is non-zero. In DFT, all contributions (Coulomb, exchange, and correlation) are described via the electron density with the (physically incorrect) consequence that there are no corresponding perturbed contributions in the case of magnetic perturbations. As there is no perturbed contribution, there is no coupUng between the perturbed orbitals and we obtain an uncoupled approach. Computationally, the uncoupled approach is easy to implement, but it needs to be emphasized that it is not correct from a theoretical viewpoint. What are the solutions to the above mentioned problems ... 

Diagonalization of F with the constraint that the first-order reduced density matrix 7 (the one-matrix) satisfies 7sr = (ar s) = (nr)Ssr with occupation numbers (n ) = 0 or 1 i.e., Tr y = N and 7 = 7) is done iteratively and converges to a single determinantal SCF approximation for the iV-electron ground state corresponding to the appropriate set of occupation numbers. 

Methods using the first-order reduced density matrix as variable can be chosen to strictly enforce the A-representability of yi, and employ the exact energy functional for all the terms except the correlation energy. The latter, however, inherently depends on 72, which in this approach must be approximated as a function of yi. One can argue that Hartree-Fock belongs to this class of methods, with imphcit neglect of the electron correlation. 

Alternative approaches to the many-electron problem, working in real space rather than in Hilbert space and with the electron density playing the major role, are provided by Bader s atoms in molecule [11, 12], which partitions the molecular space into basins associated with each atom and density-functional methods [3,13]. These latter are based on a modified Kohn-Sham form of the one-electron effective Hamiltonian, differing from the Hartree-Fock operator for the inclusion of a correlation potential. In these methods, it is possible to mimic correlated natural orbitals, as eigenvectors of the first-order reduced density operator, directly... 

Density matrices, in particular, the so-called first- and second-order reduced density matrices, are important quantities in the theoretical description of electronic structures because they contain all the essential information of the system under study. Given a set of orthonormal MOs, we define the first-order reduced density matrix D with matrix elements as the expectation value of the excitation operator E = aLa, -I- with respect to some electronic wave function Fgi,... 

Although one can extend this discussion to even higher order electron densities, p(ri) and p(ri,r2) are the most commonly used quantities in molecular quantum similarity. It is worth noting that often no order is mentioned in publications when considering electron density. It is then commonly accepted that one then refers to the first-order density. In the remainder of this chapter, we assume that the first-order density is used, except when an order is mentioned explicitly. An extremely important property of p(ri) is its positive definite nature, which may seem like a simple consequence of its probability nature, but this point can hardly be overemphasized for its application in quantum similarity, as will be shown. 

He goes on to show that for the description of the energy it is sufficient to know the second-order reduced density matrix F(jc, jc2 i 2)- This was a favorite subject when he was lecturing and led to speculations about the possibility to compute the second-order density matrix directly, and discussions of the so called A/ -represent-ability problem. In spite of several attempts, this way of attacking the quantum chemical many-particle problem has so far been unsuccessful. Of special interest was the first-order reduced density matrix, 7( 1 jC ), which when expanded in a complete one-electron basis, ij/, is obtained as... 

The probability density of electrons p x) in a quantum-mechanical system is given by the diagonal element of the first-order reduced-density matrix, (the superscript s indicates... 

Within the conventional density-functional formalism one seeks an expression for the total electronic energy as a functional of the electron density of Eq. (16). The electron density of Eq. (16) can be considered the diagonal elements of the first-order reduced density matrix... 

In order to define natural orbitals, we now consider the first-order reduced density matrix of an iV-electron system. Given a normalized wave function, O, then 0(xi,..., x y), Xjy) dx dxjy is the probability... 

The matrix y(xi, X ), which depends on two continuous indices, is called the first-order reduced density matrix or alternatively, the one-electron re-... 

D and are the unperturbed and first-order perturbed density matrices H is the one-electron Hamiltonian integral... 

You see, for the molecule to be strictly non-polar, the zero-order electron density distribution must be symmetric about z = 0, whereas is antisymmetric. Therefore, the integrand i/ aI Za is antisymmetric, and it integrates to zero (Fig. 10.15). To first order in perturbation theory, there is no dispersion force. 

Employing Eqs. (35)-(37), the first-order electronic current density induced by the external magnetic field can be written as a sum of paramagnetic and diamagnetic contributions. 

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