Scaled ground-state wavefunction

Hohenberg-Kohn orbit. Clearly, within the application of local-scaling transformations to any initial wavefunction leads to the exact ground-state wavefunction as well as to the exact ground-state density. 

We studied the behavior of various approximate EOM models based on the CCSDT-3 ground state wavefunction in IP, EA and EE calculations. We assumed that the approximate models should exhibit a lower scaling that the full method, i.e., for IP and EA problems and rP for the EE case, so the bottom line for all approaches is an elimination of the Rj, operator from the i 3 equation in the EOM scheme, cf. equation (20). 

There are many ways to compute the ground-state wavefunction Tq, such as configuration-interaction techniques and coupled-cluster methods, widely studied by the ab-initio theoretical-chemistry community. These methods however show a unfavorable scaling with the number of electrons and still cannot be applied to model large systems of increasing interest (e.g. in nano-bio-science). The main limitation is that these methods focus on the many-electron wavefunction T which contains N variables. 

Figure 5.12 Radial wavefunctions P /r) with n = 3 for neutral chlorine (Cl I), singly-ionized argon (Aril), and doubly-ionized potassium (Kill) which demonstrate the collapse of the 3d orbital (all these systems have the electron configuration ls22s22p63s23p5 in the ground state while the results apply to the. .. 3s23p4( D)3d 2S states). Note that the horizontal scales are slightly different, but the distance from the origin to (3s + 3p)/2 is the same in all three cases. It is the average position 3s and , i.e., the 3d orbital is clearly uncollapsed. The opposite is true for Kill where the 3d orbital has clearly collapsed. The 3d orbital of Aril falls between these two cases. From [SHa83].

In this section, we briefly discuss some of the electronic structure methods which have been used in the calculations of the PE functions which are discussed in the following sections. There are variety of ab initio electronic structure methods which can be used for the calculation of the PE surface of the electronic ground state. Most widely used are Hartree-Fock (HF) based methods. In this approach, the electronic wavefunction of a closed-shell system is described by a determinant composed of restricted one-electron spin orbitals. The unrestricted HF (UHF) method can handle also open-shell electronic systems. The limitation of HF based methods is that they do not account for electron correlation effects. For the electronic ground state of closed-shell systems, electron correlation effects can be accounted for relatively easily by second-order Mpller-Plesset perturbation theory (MP2). In modern implementations of MP2, linear scaling with the size of the system has been achieved. It is thus possible to treat quite large molecules and clusters at this level of theory. 

In an adiabatic regime where nuclear motion is effectively decoupled from electronic motion (due to their differences in time scale) the full Hamiltonian could be broken into two parts, namely the electronic part and the nuclear part. Accordingly, the electrons can be described by a wavefunction which by itself obeys the Schrbdinger equation. Considering the ground state we have then... 

The EMM approach couples the large scale sensitivity of moments to the localized demands of the signature property of the wavefunction. It does this through a multiscale procedure that tests for compliance with positivity (in the case of the ground state) over smaller and smaller scales, as the degree of the sampling polynomial, Vc, increases. 

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