Wave function frozen orbital

In response to perturbation, essentially all parameters that enter vary. Here we consider only the variation in the cluster amplitudes (the reference wave function and orbitals therein are frozen), i.e.,... 

The Born-Oppenheimer approximation (see Section 6.1) treats the motion of electrons and nuclei on very different time scales. The electrons, being much less massive, move much more rapidly than the nuclei in a molecule. In the Born-Oppenheimer approximation, we consider the nuclei to be frozen at particular locations and calculate the electronic energy levels and wave functions (molecular orbitals) for the rapidly moving electrons (see Chapter 6). We find that the allowed... 

It is not possible to use normal AO basis sets in relativistic calculations The relativistic contraction of the inner shells makes it necessary to design new basis sets to account for this effect. Specially designed basis sets have therefore been constructed using the DKH Flamiltonian. These basis sets are of the atomic natural orbital (ANO) type and are constructed such that semi-core electrons can also be correlated. They have been given the name ANO-RCC (relativistic with core correlation) and cover all atoms of the Periodic Table.36-38 They have been used in most applications presented in this review. ANO-RCC are all-electron basis sets. Deep core orbitals are described by a minimal basis set and are kept frozen in the wave function calculations. The extra cost compared with using effective core potentials (ECPs) is therefore limited. ECPs, however, have been used in some studies, and more details will be given in connection with the specific application. The ANO-RCC basis sets can be downloaded from the home page of the MOLCAS quantum chemistry software (http //www.teokem.lu.se/molcas). 

Ab initio calculations were carried out for all the low-lying non-Rydberg states of the systems N2, 02, NO, Of, and NO+. In N2, for example, there are 102 molecular states that result from nitrogen atoms in the lowest 4S, 2D, and 2P states. These states were all uniformly described using VCI wave functions constructed as described in Section II. Minimum basis, double-f basis and double-f-plus-polarization basis sets were employed for these studies. For the minimum basis-set calculations, which were always carried out first, the VCI wave functions represent full Cl projections with the constraint that the K shells were kept frozen for all states. However, no constraint on the 2og and 2ou orbitals was made since a Cl among these orbitals is necessary to ensure proper description of the hole states in these molecules, such as C3n of N2. The calculations all have the property of asymptotically connecting with the correct atomic states. This computational method has previously been applied, with reliable results, to both closed- and open-shell systems.6 9 11... 

The most accurate theoretical results for positronium formation in positron-helium collisions in the energy range 20-150 eV are probably those of Campbell et al. (1998a), who used the coupled-state method with the lowest three positronium states and 24 helium states, each of which was represented by an uncorrelated frozen orbital wave function... 

The basic assumption in the derivation, one generally considered valid for the time scale involved in photoemission, is the sudden approximation that is, in the wave function, i(N— 1), for the (N—l) electron ion immediately after photoemission, all the (N— 1) orbitals, except of course that of the missing ith electron, are frozen in the forms they had in the initial N electron state Wo(N). Now this i is not an eigenfunction of the actual (N—l) electron system, but it may be expanded in a complete set of such eigenfunctions, Wn(N— 1),... 

The A -shell x-ray emission rates of molecules have been calculated with the DV-Xa method. The x-ray transition probabilites are evaluated in the dipole approximation by the DV-integration method using molecular wave functions. The validity of the DV-integration method is tested. The calculated values in the relaxed-orbital approximation are compared with those of the frozen-orbital approximation and the transition-state method. The contributions from the interatomic transitions are estimated. The chemical effect on the KP/Ka ratios for 3d elements is calculated and compared with the experimental data. The excitation mode dependence on the Kp/Ka ratios for 3d elements is discussed. 

Most theoretical calculations of x-ray emission rates in atoms and molecules have been performed in the frozen-orbital (FR) approximation, where the same atomic or molecular potential is used before and after the transition. It is usual to use the ground state configuration for this purpose. This approximation is convenient because we need only one atomic or molecular calculations and the wave functions for the initial and final states are orthogonal. However, the presence of vacancy is not taken into consideration. On the other hand, we have shown that the TS method is useful to predict x-ray transition energies, but is not so good approximation to the absolute x-ray transition probabilities [39]. 

We did some preliminary calculations for BCu (which still could be treated with rather large active spaces) with different partitioning of the orbital space in CAS calculations. The notation is (frozen(inactive active n el) for orbital subspaces and n correlated electrons in the active space. The C2v symmetry was used in all computations. For most distances the wave function has definitively a two configuration form. The smallest active space considered is (0000 9331 2000 2 el) in the CASSCF calculation while in the subsequent CASPT2 calculation we used the (6220 3111 2000 2 el) space. The best would be to choose as the active space the valence orbitals of boron (2110) and the 3d,4s and the correlating 4d shell for Cu (5222). 

The method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

Consider a system with only two levels of a given symmetry for example, the hydrogen atom with only two orbitals in the orbital basis or the excitations of beryllium with a. 2s Ip basis in the frozen core (1j ) approximation. In the first example, the b—>2i excitation operator and its adjoint are the only allowed basis operators, and in the second, the 2 — 2p ( S) double excitation and its adjoint are the only basis operators of this symmetry. The two basis configurations for the system are written as 1 4) and 15 >. The normalized ground-state wave function (either exact within this basis or an approxmate one) is... 

The contracted labelling describes the fact that the expansion coefficients of the correlated wave functions used as basis set in the second-step are kept frozen in the diagonalisation of the spin-orbit matrix, making reference to the contraction coefficients of atomic basis sets used in the SCF step. 

AFfrz separated from the density relaxation terms (AEpoi and AEct) but also allows a clean separation of electrostatic and Pauli repulsion terms. Similar intermediate states in wave function-based EDA approaches are represented by the HL antisymmetrization of two fragments wave functions, [which is] necessary because molecular orbitals from different fragments are not orthogonal. This antisymmetrized wave function, however, deforms the frozen density [12] that is to say, its density does not correspond to the sum of fragments densities. Such ambiguity makes it difficult to separate electrostatic and Pauli repulsion terms in other EDA approaches. In addition, a one-step antisymmetrization of the wave functions means its energy is not variational. 

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