Particle wavefunctions

The total wavefunction r2,. . ., r is written as a product of single-particle functions (Hartree approximation). The various integrals are evaluated in tire saddle point approximation. A simple Gaussian fomr for tire trial one-particle wavefunction... 

To ensure this, the-many-body wavefunction can be written as a Slater determinant of one particle wavefunctions - this is the Hartree Fock method. The drawbacks of this method are that it is computationally demanding and does not include the many-body correlation effects. 

HF (HF) theory is based on the idea that one takes an antisymmetrized product wavefunction and uses the variational principle to obtain the best possible approximation to the A -particle wavefunction that cannot be represented by such a single determinant. Thus, one inserts the single determinant into the Rayleigh-Ritz functional and performs a constraint variation of the orbitals. The results of the variational process are the famous HF equations that are satisfied by each of the orbitals ... 

Both objects are much less complicated than the total A -particle wavefunction itself, since they only depend on three spatial variables. The electron density is manifestly positive (or zero) everywhere in space while the spin-density can be positive or negative. If, by convention, there are more spin-up than spin-down electrons, the positive part of the spin-density will prevail and there will usually be only small regions of negative spin-density that arise from spin-polarization. This spin-polarization is physically important and is already included in the UHF method but not in the ROHF method that, by construction, can only describe the... 

As a simple example of a QM/MM Car-Parinello study, we present here results from a mixed simulation of the zwitterionic form of Gly-Ala dipeptide in aqueous solution [12]. In this case, the dipeptide itself was described at the DFT (BLYP [88, 89 a]) level in a classical solvent of SPC water molecules [89b]. The quantum solute was placed in a periodically repeated simple cubic box of edge 21 au and the one-particle wavefunctions were expanded in plane waves up to a kinetic energy cutoff of 70 Ry. After initial equilibration, a simulation at 300 K was performed for 10 ps. 

Kohn and Sham provided a further contribution to make the DFT approach useful for practical calculations, by introducing the concept of fictitious non-interacting electrons with the same density as the true interacting electrons [8]. Non-interacting electrons are described by orthonormal single-particle wavefunctions y/i (r) and their density is given by ... 

Although we have employed Zumbach and Maschke s technique for producing the transformation vectors and these vectors have been used in a wider context for the purpose of finding the transformation that connects two arbitrary densities pi(r) and pzir), associated to iV-particle wavefunctions 4>ie Cfi and respectively. In this important sense, the procedure... 

Let us consider an N-particle wavefunction pj( i,..., r y) (where for simplicity we disregard spin) associated with the one-particle density Pi(r), and let us obtain the Fourier transform of this wavefunction by means of ... 

Because of the one-to-one correspondence between one-particle densities, p(F)e>i, and N-particle wavefunctions, within each one of the... 

An A-representable RDM is also defined to be S-representable if it derives from an A-particle wavefunction or an ensemble of A-particle wavefunctions with a definite spin quantum number 5 [57]. By definition, an 5-representable two-electron RDM yields the correct expectation value... 

Nakatsuji [37] in 1976 first proved that with the assumption of N-representability [3] a 2-RDM and a 4-RDM will satisfy the CSE if and only if they correspond to an A-particle wavefunction that satishes the corresponding Schrodinger equation. Just as the Schrodinger equation describes the relationship between the iV-particle Hamiltonian and its wavefunction (or density matrix D), the CSE connects the two-particle reduced Hamiltonian and the 2-RDM. However, because the CSE depends on not only the 2-RDM but also the 3- and 4-RDMs, it cannot be solved for the 2-RDM without additional constraints. Two additional types of constraints are required (i) formulas for building the 3- and 4-RDMs from the 2-RDM by a process known as reconstruction, and (ii) constraints on the A-representability of the 2-RDM, which are applied in a process known as purification. 

By Eq. (6) the sum on the right-hand side of the above equation is equal to the energy E, and from Eq. (2) we realize that the sums on the left-hand side are just Hamiltonian operators in the second-quantized notation. Hence, when the 2-RDM corresponds to an A -particle wavefunction i//, Eq. (12) implies Eq. (13), and the proof of Nakatsuji s theorem is accomplished. Because the Hamiltonian is dehned in second quantization, the proof of Nakatsuji s theorem is also valid when the one-particle basis set is incomplete. Recall that the SE with a second-quantized Hamiltonian corresponds to a Hamiltonian eigenvalue equation with the given one-particle basis. Unlike the SE, the CSE only requires the 2- and 4-RDMs in the given one-particle basis rather than the full A -particle wavefunction. While Nakatsuji s theorem holds for the 2,4-CSE, it is not valid for the 1,3-CSE. This foreshadows the advantage of reconstructing from the 2-RDM instead of the 1-RDM, which we will discuss in the context of Rosina s theorem. 

In the last few years, the improvements in computer hardware and software have allowed the simulation of molecules and materials with an increasing number of atoms. However, the most accurate electronic structure methods based on N-particle wavefunctions, for example, the configuration interaction (Cl) method or the coupled-cluster (CC) method, are computationally too expensive to be applied to large systems. There is a great need for treatments of electron correlation that scale favorably with the number of electrons. 

It seems that there is no A-representabihty problem since the conditions that ensure that a one-particle density comes from an A-particle wavefunction are well known [5]. Here, the obstacle is the constmction of the functional E [p] capable of describing a quantum mechanical A-electron system. This functional A-representabUity is still related to the A-representabUity problem of the 2-RDM. Many currently available functionals are not A-representable [6]. Consequently, the energies produced by these functionals can lie below the exact value. Even though these energy values may lie quite close to the exact ones, they do not guarantee, however, that the calculations are accurate. 

A direction for improving DPT lies in the development of a functional theory based on the one-particle reduced density matrix (1-RDM) D rather than on the one-electron density p. Like 2-RDM, the 1-RDM is a much simpler object than the A-particle wavefunction, but the ensemble A-representability conditions that have to be imposed on variations of are well known [1]. The existence [10] and properties [11] of the total energy functional of the 1-RDM are well established. Its development may be greatly aided by imposition of multiple constraints that are more strict and abundant than their DPT counterparts [12, 13]. 

A new algorithm is presented for the calculation of energy levels and their associated second-order density matrices, which aims to produce the exact energy as in full configuration interaction but without the V-particle wavefunction. 

The smallest value of cr for which the determinant B vanishes is such that there exists an A-particle wavefunction, ij/, for which B ij/ = B P = 0 and therefore (/ + aH) jj = 0. It follows from this that... 

One can impose additional spatial confinement on a Debye plasma such that the potential energy vanishes at the boundary of a given sphere of radius R. For the strongly coupled system, one can assume that no electron current passes through the boundary surface and the wavefunction must vanish at the Wigner-Seitz boundary R [154], Under such conditions, the radial one-particle wavefunction ir(r) satisfies... 

The MO concept is directly related to an approximate wavefunction consisting of a Slater determinant of occupied one-particle wavefunctions, or molecular orbitals. The Hartree-Fock orbitals are by definition the ones that minimize the expectation value of the Hamiltonian for this Slater determinant. They are usually considered to be the best orbitals, although it should not be forgotten that they are only optimal in the sense of energy minimization. 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

Renaming the electron numbers shows that the first and fourth terms in the matrix element M i(M1,ms) and also the second and third terms are identical, and they can be combined. As a next step one calculates the action of the photon operator on the single-particle wavefunctions. Omitting for simplicity the wavefunction... 

The correlated wavefunction which incorporates ISCI follows in analogy to equ. (1.25a) as an expansion into independent-particle wavefunctions for the ground state and contributions from virtual two-electron excitations ... 

All three forms of the dipole matrix element are equivalent because they can be transformed into each other. However, this equivalence is valid only for exact initial- and final-state wavefunctions. Since the Coulomb interaction between the electrons is responsible for many-body effects (except in the hydrogen atom), and the many-body problem can only be solved approximately, the three different forms of the matrix element will, in general, yield different results. The reason for this can be seen by comparing for the individual matrix elements how the transition operator weights the radial parts R r) and Rf(r) of the single-particle wavefunction differently ... 

If we are interested in the ground-state electronic properties of a molecule or solid with a given set of nuclear coordinates we should seek the solution to the Schrodinger equation which corresponds to the lowest electronic energy of the system. However, the inter-electronic interactions in eq. (2.2) are such that this differential equation is non-separable. It is therefore impossible to obtain the exact solution to the full many-body problem. In order to proceed, it is necessary to introduce approximation in this equation. Two types of approximations can be separated, namely, approximations of the wavefunction, VF, from a true many-particle wavefunction to, in most... 

When these transformations are applied to an initial arbitrary 7 7-particle wavefunction, Ir5, one obtains a density-dependent transformed wavefunction, 15 ). An energy functional containing both the one-particle density and the initial arbitrary wavefunction arises when the (density-dependent) transformed... 

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