One-electron basis

Let us assume that a complete set of the orthonormalized functions y n(r) of the spatial coordinates r is known. They form a basis in the space L = L2(R3) of the square integrable functions of r, known in this context as orbitals. The completeness condition means that the following holds  

It is much easier to introduce the complete basis in the space of functions depending on the spin variable of one electron. Allowable values of the spin projection s (in the units of K) are 1/2. Corresponding functions have the form  

This explains the commonly used terms according to which a electrons are those with spin-up whereas / -electrons have spin-down . The orthonormality and completeness of the set of functions a(s) and f3(s) can be checked easily. 

Upon integration, the overlap integral is found to take the form 

From the bonding and antibonding MOs (5.2.3) and (5.2.4), we may generate a total of six determinants by distributing the two electrons among the four spin orbitals in all possible ways allowed for by the Pauli principle. In our calculations, however, we shall employ a basis of CSFs rather than determinants. As discussed in Section 2.5, the CSFs are spin-adapted linear combinations of determinants belonging to the same orbital configuration. 

Applying to the vacuum state the two-body creation operators introduced in Section 2.3.3, we obtain two closed-shell CSFs of symmetry (the bonding and antibonding configurations) 

Only states of the same spin and space symmetries are coupled by the totally symmetric electronic Hamiltonian. Thus, the only CSFs that will mix in our calculations are those of gerade symmetry. An arbitrary, normalized state of this symmetry may be written in the following manner 

In the remainder of Section 5.2, we shall investigate the description of the H2 electronic system made possible by the six CSFs generated from the minimal atomic basis. In particular, we shall examine and compare the electronic energies and the density functions associated with the different states generated in this basis. 

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Because the interelectronic cusp is difficult to describe well with one-electron basis functions, pair correlation energies converge much more slowly (as N" ) than SCF energies (which converge as f ). This fact makes the use of CBS extrapolations of the correlation energy very beneficial in terms of both accuracy and computational cost. 

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (4.1) can similarly be considered as describing the total wave function in a coordinate system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation). 

Here Z is the charge of the projectile with velocity v. In order to calculate stopping powers for atomic and molecular targets with reliability, however, one must choose a one-electron basis set appropriate for calculation of the generalized oscillator strength distribution (GOSD). The development of reasonable criteria for the choice of a reliable basis for such calculations is the concern of this paper. 

A standard method of improving on the Hartree-Fock description is the coupled-cluster approach [12, 13]. In this approach, the wavefunction CC) is written as an exponential of a cluster operator T working on the Hartree-Fock state HF), generating a linear combination of all possible determinants that may be constructed in a given one-electron basis,... 

The Correlation-Consistent Hierarchy of One-Electron Basis Sets... 

The quality of quantum-chemical calculations depends not only on the chosen n-electron model but also critically on the flexibility of the one-electron basis set in terms of which the MOs are expanded. Obviously, it is possible to choose basis sets in many different ways. For highly accurate, systematic studies of molecular systems, it becomes important to have a well-defined procedure for generating a sequence of basis sets of increasing flexibility. A popular hierarchy of basis functions are the correlation-consistent basis sets of Dunning and coworkers [15-17], We shall use two varieties of these sets the cc-pVXZ (correlation-consistent polarized-valence X-tuple-zeta) and cc-pCVXZ (correlation-consistent polarized core-valence X-tuple-zeta) basis sets see Table 1.1. 

In our discussion so far, we have used electronic energies that are assumed to represent calculations carried out in an infinite basis of one-particle functions (the basis-set limit). In practice, finite basis sets are used as we shall see, the truncation of the one-electron basis is a serious problem that may lead to large errors in the calculations. 

The slow convergence of the correlation energy with the one-electron basis set expansion has provided the motivation for several attempts to extrapolate to the complete basis set limit [6-13], Such extrapolations require a well defined sequence of basis sets and a model for the convergence of the resulting sequence of approximations to the... 

The development of these explicit-rjj methods has yielded a database of benchmark results for small polyatomic molecules. These calculations are listed as MP2-R12 and CCSD(T)-R12 in our tables. We have selected the version called MP2-R12/A as a benchmark reference for our study of the convergence to the MP2 limit. This is the version that Klopper et al. found to agree best with our interference effect. The close agreement with extrapolations of one-electron basis set expansions justifies this choice. 

Fig. 13.4 Logarithm of error(Eh) in the configuration interaction energy for the ground state of the helium atom as a function of maximum orbital quantum number, L, of the one-electron basis functions. The data were obtained in an...

This is what we consistently find by monitoring EPR and Mossbaner spectra (cooperation with Prof. E. Miinck, Pittsburgh) of the Nia-S-CO, Nia-C " and Nia-SR states the A. vinosum enzyme produced by H2 and/or CO at pH 8. A possible explanation is to assume that the individual enzyme molecules can exchange electrons. The best suited place for this is via the distal [4Fe-4S] cluster which is located on the surface of the protein. Such an exchange would occur on a one-electron basis and wonld be much slower (depends on the collision rate of the 90kDa enzyme) than the reaction with H2 (which is extremely fast and depends on the rate of diffusion of H2 into the enzyme (Pershad et al. 1999)). Suppose that the NC-C state is initially formed with one Ee-S cluster in the oxidized state ... 

From the results presented in Figs 3 and 4 and unpublished results we can conclude, that for sufficiently large one-electron basis set, i.e., at least of aug-... 

We report MCSCF calculations of the dipole and quadmpole polarizability tensor radial functions of LiH and HF for internuclear distance reaching from almost the unified atom to the dissociation limit. Large one-electron basis sets and MCSCF wavefunctions of the CAS type with large active spaces were employed in the calculations. 

We have investigated the dependence of the polarizability radial functions on the one-electron basis set and the size of the CAS. We find that it is necessary to use basis sets of at least daug-cc-pVQZ quahty for these properties. With respect to the size of the active space we observe very large differences between the results obtained with a (7331) and (8441) active space, but only small changes on increasing the active space to (9552). We conclude, therefore, that we recover the important correlation contributions with a lOO CAS wavefunction. Furthermore, we found that the results of the calculations can be reproduced... 

These one-electron basis functions, 4>, constitute the basis set. When the basis functions represent the atomic orbitals for the atoms in the molecule, eq. 3.4 corresponds to a linear combination of atomic orbitals (LCAO) approximation. 

Following Ref. [5] the T1 condition is obtained by considering an operator A = Y ij gij,kaiajak, where the gij k are arbitrary real or complex coefhcients totally antisymmetric in the three indices. (We view g as a vector of dimension (0, where r is the size of the one-electron basis.) The contractions (t / A+A t /) and (t / AA+ t /) both involve the 3-RDM, but with opposite sign, and so the nonnegativity of (tk 4 4 -f AA I ) for all three-index functions g provides a representability condition involving only the 1-RDM and 2-RDM. In exphcit form the condition is of semidefinite form, 0 T, where the Hermitian matrix T is... 

In what follows, the number N of particles is assumed to be a constant. The one-electron basis is assumed to be finite and formed by 2JC orthonormal spin orbitals denoted by the italic letters i,j, k,l... or, when the spin is considered explicitly by ia or. ... 

A -representability conditions [28]. Let us start this description by focusing on the RDM s properties, which may be deduced from their definition as expectation values of density fermion operators. Thus the ROMs are Hermitian, are positive semidefinite, and contract to finite values that depend on the number of electrons, N, and in the case of the HRDMs on the size of the one-electron basis of representation, 2/C. Thus... 

We keep in mind the option to extend the one-electron basis, in terms of which H is defined, to the limit where it becomes complete [10]. 

It is convenient to use a one-electron basis of natural spin orbitals (NSOs) in terms of which y is diagonal. 

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