Complex Kohn variational method

Previous studies [37, 38] of the doubly excited autoionizing states of H2 show that the lowest resonance is dominated by the s and d partial waves. Therefore, we use three Is-cSTO-fVGs and three 3d-cSTO-A/Gs. As discussed in Introduction, the selection of complex orbital exponents is not an easy task at all, and in this work, we attempt to propose a systematic way to select the orbital exponents for cSTOs. A hint for the selection can be obtained by relating the CBF method to the complex Kohn variation method [22, 39]. In the latter method, the outgoing continuum wave function is represented as a linear combination of basis functions and a few non functions satisfying the outgoing asymptotic behavior. In the CBF method, only functions are used thus, additional basis functions need to... 

In this article we focus on one of these approaches, the complex Kohn (CK) method, both because it is a singularly successful example of the variational methods and because it provides a particularly clear view of the central problem of electron scattering, namely the consistent treatment of electronic correlation in the target molecule and correlation involving the scattered electrons. The CK method has its origins in Kohs s 1948 paper on variational principles, and some connections between this method and various other approaches have been discussed by McCurdy, Rescigno, and Schneider. ... 

Kohn variational theory 8.2.3 The complex Kohn method... 

For variational methods, such as Hartree-Fock (HF), multi-configurational self-consistent field (MCSCF), and Kohn-Sham density functional theory (KS-DFT), the initial values of the parameters are equal to zero and 0) thus corresponds to the reference state in the absence of the perturbation. The A operators are the non-redundant state-transfer or orbital-transfer operators, and carries no time-dependence (the sole time-dependence lies in the complex A parameters). Furthermore, the operator A (t)A is anti-Hermitian, and tlie exponential operator is thus explicitly unitary so that the norm of the reference state is preserved. Perturbation theory is invoked in order to solve for the time-dependence of the parameters, and we expand the parameters in orders of the perturbation... 

Because the convenience of the one-electron formalism is retained, DFT methods can easily take into account the scalar relativistic effects and spin-orbit effects, via either perturbation or variational methods. The retention of the one-electron picture provides a convenient means of analyzing the effects of relativity on specific orbitals of a molecule. Spin-unrestricted Hartree-Fock (UHF) calculations usually suffer from spin contamination, particularly in systems that have low-lying excited states (such as metal-containing systems). By contrast, in spin-unrestricted Kohn-Sham (UKS) DFT calculations the spin-contamination problem is generally less significant for many open-shell systems (39). For example, for transition metal methyl complexes, the deviation of the calculated UKS expectation values S (S = spin angular momentum operator) from the contamination-free theoretical values are all less than 5% (32). 

Equation (15) is the key equation of the Kohn variational principle for the -matrix (21). For small problems, when the spectral representation of ft can be obtained, both methods are essentially equivalent. If the linear equations are to be solved iteratively, the present method, Eq. (14), effectively requires to solve half the number of sets of simultaneous linear equations as the basis and xT can chosen real making Eq. (14) real while (15) remains complex. 

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