Frozen atomic orbitals

The calculation performed for the metastable N (ls2s) + He system has necessitated somewhat larger Cl spaces (200-250 determinants) in order to reach the same perturbation threshold ri = 0.01, the la molecular orbital being not frozen for this calculation.The basis of atomic orbitals has been also expanded to a 10s6p3d basis of gaussian functions for nitrogen reoptimized on N (ls ) for the s exponents and on N (ls 2p) for the p exponents and added of one s and one p diffuse functions [22]. For such excited states,... 

All the above methods are somehow based on an orbital hypothesis. In fact, in the multipolar model, the core is typically frozen to the isolated atom orbital expansion, taken from Roothan Hartree Fock calculations (or similar [80]). Although the higher multipoles are not constrained to an orbital model, the radial functions are typically taken from best single C exponents used to describe the valence orbitals of a given atom [81]. Even tighter is the link to the orbital approach in XRCW, XAO, or VOM as described above. Obviously, an orbital assumption is not at all mandatory and other methods have been developed, for example those based on the Maximum Entropy Method (MEM) [82-86] where the constraints/ restraints come from statistical considerations. 

Figure 6.2 Auger decay width of (2p n) Mg+-H+ as a function of the Mg-proton distance, R.z is Mg-proton axis. Diamonds and solid line Fano-ADC(2)x calculation with atomic orbital basis centered both on Mg and on the proton circles and long-dashed line Fano-ADC(2)x calculation with atomic orbital basis centered only on Mg stars and short-dashed line Fano-ADC(2)x calculation for (2p 1) Mg+ alone, with atomic orbital basis centered both on Mg and at the distance R along the z-axis, showing the so-called basis set superposition error (BSSE) triangles and dashed-dotted line Fano-ADC(2)x calculation with atomic orbital basis centered on Mg only, with the 3s orbital of Mg being frozen at its shape at R = 6.5A. The inset shows the low-r part of the plot on logarithmic scale. See Ref. [35] for the details of the computation.

It is quite common in correlated methods (including many-body perturbation theory, coupled-cluster, etc., as well as configuration interaction) to invoke the frozen core approximation, whereby the lowest-lying molecular orbitals, occupied by the inner-shell electrons, are constrained to remain doubly-occupied in all configurations. The frozen core for atoms lithium to neon typically consists of the Is atomic orbital, while that for atoms sodium to argon consists of the atomic orbitals Is, 2s, 2px, 2py and 2pz. The frozen molecular orbitals are those made primarily from these inner-shell atomic orbitals. 

Virtually all non-trivial collision theories are based on the impact-parameter method and on the independent-electron model, where one active electron moves under the influence of the combined field of the nuclei and the remaining electrons frozen in their initial state. Most theories additionally rely on much more serious assumptions as, e.g., adiabatic or sudden electronic transitions, perturbative or even classical projectile/electron interactions. All these assumptions are circumvented in this work by solving the time-dependent Schrodinger equation numerically exact using the atomic-orbital coupled-channel (AO) method. This non-perturbative method provides full information of the basic single-electron mechanisms such as target excitation and ionization, electron capture and projectile excitation and ionization. Since the complex populations amplitudes are available for all important states as a function of time at any given impact parameter, practically all experimentally observable quantities may be computed. 

A similar approach can be applied for the Y atom insertion into the C-H bond of alkenes and other alkanes. Our calculation by the Cl method in a 6-311-H-G(2d,2p) basis set with a complete active space for 8 electrons in 8 orbitals (Is orbital of carbon atom is frozen) predicts that the vertical S-T excitation energy in methane is around 11 eV (11.37 eV or 262 kcal/mol). Following the above approximation it is equal -2 Jch- From Eq. (9) the activation energy for the yttrium atom insertion reaction, Eq. (1) M=Y, should be 17.6 kcal/mol. This simple estimation is in a good agreement with very accurate ab initio calculations, Ea = 20.7 kcal/mol [15]. 

The use of frozen orbitals, such as the bond orbitals connecting the quantum to the classical part of the system, can be extended to nonempirical quantum methods such as ab initio Hartree -Fock, post Hartree Fock, or DFT. In these cases, the overlap between atomic orbitals is taken into account and the orthogonality conditions are more difficult to fulfill. The mathematical formulation of the method has been developed in the original papers [26 28] and the process can be summarized as follows. 

Project the N atomic orbitals of the basis set out of the subspace defined by the frozen orbitals. The result is a set of Nfunctions orthogonal to the frozen orbitals, but this set is not linearly independent because there exist L additional linear combinations orthogonal to them. 

Effective potentials also depend on the type of basis set used, hi atomic orbital calculations, they are sometimes referred to as frozen-core potentials. In most cases, only the highest-energy s, p and d electrons are included in the calculation. In plane-wave calculations, effective potentials are known as pseudopotentials They come in different varieties soft or ultrasoft pseudopotentials need only a relatively low energy cut-off as they involve a larger atomic core. ... 

So far, we have mentioned methods that produce all-electron diabatic wavefunctions and corresponding Hamiltonian matrix elements. There are two other classes of methods which simplify the quantum problem by focusing on the wavefunction of the transferred charge such as methods making use of the frozen core approximation Fragment Orbital methods (FO), and methods that assume the charge to be localized on single atomic orbitals [50]. In this work, we will also treat these computationally low-cost methods. 

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