Many-electron wave functions, electronic structure calculations

The minimization of this functional presents a problem which for many component mixtures can be quite timeconsuming if the truly optimal form of the interface and free energy is to be found. One may use an iterative method of solution much like the famous scheme used to solve for the Hartree-Fock wave function in electronic structure calculations [4]. An alternative, much to be preferred when sufficiently accurate, is to use a simple parametrized form for the particle densities through the interface and then determine the optimal values of these parameters. The simplest possible scheme is, of course, to take the profile to be a step function. 

When a tunneling calculation is undertaken, many simplifications render the task easier than a complete transport calculation such as the one of [32]. Let us take the formulation by Caroli et al. [16] using the change induced by the vibration in the spectral function of the lead. In this description, the current and thus the conductance are proportional to the density of states (spectral function) of the leads (here tip and substrate). This is tantamount to using some perturbational scheme on the electron transmission amplitude between tip and substrate. This is what Bardeen s transfer Hamiltonian achieves. The main advantage of this approximation is that one can use the electronic structure calculated by some standard way, for example plane-wave codes, and use perturbation theory to account for the inelastic effect. In [33], a careful description of the Bardeen approximation in the context of inelastic tunneling is given, and how the equivalent of Tersoff and Hamann theory [34,35] of the STM is obtained in the inelastic case. 

Electronic structure calculations may be carried out at many levels, differing in cost, accuracy, and reliability. At the simplest level, molecular mechanics (this volume, Chapter 1) may be used to model a wide range of systems at low cost, relying on large sets of adjustable parameters. Next, at the semiempirical level (this volume, Chapter 2), the techniques of quantum mechanics are used, but the computational cost is reduced by extensive use of empirical parameters. Finally, at the most complex level, a rigorous quantum mechanical treatment of electronic structure is provided by nonempirical, wave function-based quantum chemical methods [1] and by density functional theory (DFT) (this volume, Chapter 4). Although not treated here, other less standard techniques such as quantum Monte Carlo (QMC) have also been developed for the electronic structure problem (for these, we refer to the specialist literature, Refs. 5-7). 

If a vector space representation of electronic states is chosen, that is, a basis-set expansion, two types of basis sets are needed. One for the many-electron states and one for the one-particle states. For the latter, two choices became popular, the molecular orbital (MO) [9] and valence bond (VB) [10] expansions. Both influenced the understanding and interpretation of the chemical bond. A bonding analysis can then be performed in terms of their basic quantities. Although both representations of the wave function can be transformed (at least partially) into each other [11,12], most commonly an MO analysis is employed in electronic structure calculations for practical reasons. Besides, a VB description is often limited to small atomic basis sets as (semi-)localized orbitals are required to generate the VB structures [13]. If, however, diffuse functions with large angular momenta are included in the atomic orbital basis, a VB analysis suffers from their delocalization tails. As a consequence, the application of VB methods can often be limited to organic molecules. 

Nowadays, many electronic structure codes include efficient implementations [37—41] of the Ramsey equations [42] for the calciflations of nonrelativistic spin—spin coupling constants. A vast number of publications devoted to the calculation of/-couplings can be found in the Hterature, covering different aspects such as the basis set effects [43-55], the comparison of wave function versus density functional theory (DFT) methods [56-60], or the choice of exchange-correlation functional in DFT approaches [61-68]. Excellent recent reviews of Contreras [69] andHelgaker [70] cover these particular aspects. 

The Hartree-Fock (HF) method is widely used in electronic structure calculations, which is based on the following assumptions (1) Bom-Oppenheimer approximation, (2) the many-electron Hamiltonian is replaced with an effective one-electron Hamiltonian which acts on orbitals (one-electron wave functions), (3) the Coulomb repulsion between electrons is represented in an averaged way. 

Electron structure calculations often become difficult when transition metals are involved. If the system has incomplete d shells many electron configurations contribute even to the ground state leading to non-dynamical electron correlation. Wave function-based methods with multideterminant references are required for high accuracy. Density functional theory is often successful, but no current functional is reliable for transition metals compounds in general. QMC as a wave function-based method has to use multideterminant... 

Density-functional theory (DFT) is one of the most widely used quantum mechanical approaches for calculating the structure and properties of matter on an atomic scale. It is nowadays routinely applied for calculating physical and chemical properties of molecules that are too large to be treatable by wave-function-based methods. The problem of determining the many-body wave function of a real system rapidly becomes prohibitively complex. Methods such as configuration interaction (Cl) expansions, coupled cluster (CC) techniques or Moller Plesset (MP) perturbation theory thus become harder and harder to apply. Computational complexity here is related to questions such as how many atoms there are in the molecule, how many electrons each atom contributes, how many basis functions are... 

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