Born-Handy ansatz

Kutzelnigg, W. The adiabatic approximation 1. The physical background of the Born-Handy ansatz. Mol. Phys. 1997, 90,909-16. 

The statement above is the reason why Kutzelnigg finally prefers the more pragmatic Born-Handy ansatz [6] proving its fiiU equivalency with the COM separation Handy and co-workers have never claimed to have invented the ansatz referred to here as the Born-Handy ansatz , but they certainly convinced a large audience that this ansatz is of enormous practical value, even if it has not been completely obvious why it leads to correct results. Handy and co-workers realized that the difficulties with the traditional approach come from the separation of the COM motion (and the need to define internal coordinates after this separation has been made). They therefore decided to renounce the separation. ... 

This is without doubt a significant improvement since the application of the Born-Handy ansatz as a full replacement of the COM separation is not restricted by the size of the system under investigation. However, its applicability is unfortunately limited to adiabatic systems only. The Born-Handy formulation yields only the adiabatic corrections to the B-0 results. Beyond this approach we enter the enigmatic region, which is usually denoted as the break-down of B-O approximation. Therefore our main goal is here to find an extension of the Born-Handy formula , which is valid both in the adiabatic limit as well as beyond. 

In this work we will emphasize those steps, which improves our previous approach where the COM problem was ignored. By summarizing the conversion of the Born-Handy ansatz to the CPHF compatible form [22], we will continue with a presentation of the consequences of this result for correct quantizations of the total Hamiltonian. This will be accompanied with a rederivation of the Born-Handy formula from the newly developed theory, and it will be shown to be of significance for the Jahn-Teller effect, for conductors, and for superconductors as well. 

Let us start with the Born-Handy ansatz [6] for the groundstate electron wave-function y/oC ) where represents nuclear coordinates. The adiabatic correction A o to the groundstate is expressed as a mean value of the nuclear kinetic operator Tn [22],... 

The Born-Handy ansatz [6] was verified on simple molecular systems many times in the last years [25-27], and especially interesting is the comparison of this simple pragmatic ansatz with the rigorous methods based on the separation of the centre-of-mass motion where one gets rather complicated expressions in terms of relative coordinates in a molecule-fixed frame. Kutzelnigg proved the validity of the Born-Handy ansatz by means of centre-of-mass analysis [5]. We can now ask what happens in the case of the break-down of the adiabatic approximation If the adiabatic case, beyond the B-0 approximation, is a general centre-of-mass problem. 

Simultaneous use of the canonical transformations and introduction of degrees of freedom is unbalanced [5] and just this is a reason why rigorous methods eliminate their introduction at all [3,4]. Nevertheless, if we insist on the introduction of degrees of freedom, we need to find an alternative to canonical transformations. To solve the COM problem on the many-body level therefore means to solve the compatibility problem of the many-body treatment with the Born-Handy ansatz, where the degrees of freedom are inseparable and have only a relative meaning. 

PhD thesis for the quantization of electron-vibrational Hamiltonian. Now a revised version will be presented for the complete electron-hypervibrational Hamiltonian which corrects the original version which totally failed during comparison tests with the results of the Born-Handy ansatz. 

Unfortunately Frohlich could not know that the electron-phonon interaction was not the true interaction model and applied his transformation to the wrong Hamiltonian. Since the author did not initially involved the COM problem into the consideration, he made the same mistake as Frohlich in all previous works referred to [12-20] with the aforesaid negative consequences for the later comparison tests [22] with the Born-Handy ansatz. Therefore we now present the solution which correctly incorporates all 3N degrees of freedom, unified under the conception of electron-hyperphonon interaction. 

Derivation of the Extended Born-Handy Ansatz from the General Representation... 

If we proceed from the general to the adiabatic representation with zero c coefficients we obtain exactly the Born-Handy ansatz (28.14) from the first principle derivation ... 

The main goal of this work was the implementation of the COM problem into the many-body treatment. The many years experience with the inconvenience of the direct COM separation on the molecular level and its consequent replacement with the Born-Handy ansatz as a full equivalent was taken into account. It was shown that the many-body treatment based on the electron-vibrational Hamiltonian is fundamentally inconsistent with the Born-Handy ansatz so that such a treatment can never respect the COM problem. 

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