Hamiltonian determination from electron density

Thus, the electron density already provides all the ingredients that we identified as being necessary for setting up the system specific Hamiltonian and it seems at least very plausible that in fact p( ) suffices for a complete determination of all molecular properties (of course, this does not relieve us from the task of actually solving the corresponding Schrodinger equation and all the difficulties related to this). As noted by Handy, 1994, these very simple and beautifully intuitive arguments in favor of density functional theory are attributed to E. B. Wilson. So the answer to the question posed in the caption to this section is certainly a loud and clear Yes . 

According to the Hohenberg-Kohn theorem of the density functional theory, the ground-state electron density determines all molecular properties. E. Bright Wilson [46] noticed that Kato s theorem [47,48] provides an explicit procedure for constructing the Hamiltonian of a Coulomb system from the electron density ... 

Secondly, information is obtained on the nature of the nuclei in the molecule from the cusp condition [11]. Thirdly, the Hohenberg-Kohn theorem points out that, besides determining the number of electrons, the density also determines the external potential that is present in the molecular Hamiltonian [15]. Once the number of electrons is known from Equation 16.1 and the external potential is determined by the electron density, the Hamiltonian is completely determined. Once the electronic Hamiltonian is determined, one can solve Schrodinger s equation for the wave function, subsequently determining all observable properties of the system. In fact, one can replace the whole set of molecular descriptors by the electron density, because, according to quantum mechanics, all information offered by these descriptors is also available from the electron density. 

The requirement needed to incorporate the solvent effects into a state-specific (multireference) method is fulfilled by using the effective Hamiltonian defined in Equation (1.159). The only specificity to take into account is that in order to calculate Va we have to know the density matrix of the electronic state of interest (see the contribution by Cammi for more details). Such nonlinear character of Va is generally solved through an iterative procedure [35] at each iteration the solvent-induced component of the effective Hamiltonian is computed by exploiting Equation (1.157) with the apparent charges determined from the standard ASC equation with the first order density matrix of the preceding step. At each iteration n the free energy of each state K is obtained as... 

The descriptors developed to characterize the substrate chemotypes are obtained from a mixture of molecular orbital calculations and GRID probe-pharmacophore recognition. Molecular orbital calculations to compute the substrate s electron density distribution are the first to be performed. All atom charges are determined using the AMI Hamiltonian. Then the computed charges are used to derive a 3D pharmacophore based on the molecular electrostatic potential (MEP) around the substrate molecules. 

Pi(r) = p[A. V r. to another, P2(r) = p[iV2,v2 r], both uniquely determined by the two state parameters determining the current electronic hamiltonian the overall number of electrons Nt and the external potential due to the nuclei v,(r), i= 1,2. We call such shifts in the system electronic structure the horizontal displacements [7,33] on the ground-state density surface , p[N, v r] = p0(r). A displacement from one v-representable electronic density to another gives rise to the associated change in the generalized density functional for the ground-state energy ... 

The sweeping theorem of Hohenberg and Kohn is that, like the wave function, the ground state s electron density determines all the properties of an electronic system [1]. The result is proved in three steps. First, one recalls that the number of electrons is determined from the electron density using Eq. (14). Next, one demonstrates that the external potential can be determined from the ground-state electron density. From N and v(r), we may determine the electronic Hamiltonian and solve Schro-dinger s equation for the wave function, subsequently determining all observable properties of the system. 

In the many years that have passed since the publication of the KS paper, the way that their results are usually presented, and where the emphasis is put, has of course shifted. It is now customary to stress from the outset that the KS theory introduces a system of noninteracting electrons, moving in a local potential, Vs(r). The ground-state wavefunction of the KS system - a single Slater determinant of the lowest N orbitals - will yield precisely the same electron density as the exact interacting electron system with potential v(r). So the KS Hamiltonian, Hs, is just a sum of one-electron Hamiltonians, hs, and the wavefunction of the KS system is a simple one-determinantal wavefunction,... 

The spin Hamiltonian parameters of a complex or a model system can in principle be determined from a quantum chemical calculation of the electronic structure. The Density Functional Theory (DFT) [13] method is at present popular for this purpose, since it allows fairly large systems ( 200 atoms) to be investigated. By comparing the calculated and experimental spin Hamiltonian parameters it is often possible to distinguish between different proposed models and to gain further insight into the electronic and geometric structure of the sample. 

---