Ground state energy minimization

Xi,, which can be varied. The quantity is then a funetion of these parameters (X, X2,. ..). For eaeh set of parameter values, the corresponding value of (Ai, /I2,. ..) is always greater than or equal to the true ground-state energy Eq. The value of X, X2,. ..) closest to Eq is obtained, therefore, by minimizing with respeet to eaeh of these parameters. Selecting a sufficiently large number of parameters in a well-chosen analytieal form for the trial funetion 0 yields an approximation very close to Eq. 

Coming back to the variational principle, the strategy for finding the ground state energy and wave function should be clear by now we need to minimize the functional E[ F] by searching through all acceptable N-electron wave functions. Acceptable means in this context that the trial functions must fulfill certain requirements which ensure that these func-... 

Greek indices a, p = x,y,z of the Cartesian coordinate axes is meant). Minimization of expression (2.1.1) for an arbitrary two-dimensional Bravais lattice becomes possible since the Fourier representation in q implies the reduction of the double sum over j and/ to the single sum over q. Then the ground state energy is given by... 

Taking a hint from the treatment of hehum by HyUeraas, 1 reahzed that one merely had to choose D 12,1 2 ) to minimize the above expression for fixed N and with K appropriate for any quantum system of N identical fermions to obtain the ground-state energy level. 

If the G-matrix is positive semidefinite, then the above expectation value of the G-matrix with respect to the vector of expansion coefficients must be nonnegative. Similar analysis applies to G, operators expressible with the D- or Q-matrix or any combination of D, Q, and G. Therefore variationally minimizing the ground-state energy of n (H Egl) operator, consistent with Eq. (70), as a function of the 2-positive 2-RDM cannot produce an energy less than zero. For this class of Hamiltonians, we conclude, the 2-positivity conditions on the 2-RDM are sufficient to compute the exact ground-state A-particle energy on the two-particle space. 

D. A. Mazziotti, Variational minimization of atomic and molecular ground-state energies via the two-particle reduced density matrix. Phys. Rev. A 65, 062511 (2002). 

Hohenberg and Kohn have proved generally that the total ground state energy E of a collection of electrons in the presence of an externally applied potential (e.g. the valence electrons in the presence of the periodic potential due to the cores in a lattice), when no net magnetic moment is present, depends only on the average density of electrons n(R). By this proof, n(R) becomes the fundamental variable of the system (as it is in the Thomas-Fermi theory ). Variational minimization of the most general form of E, with respect to n lends to the Hartree-Fock equations formalism. 

The effective Hamiltonian depends on pararemeters Cia- The optimal choice can be obtained by minimizing the ground state energy of H ff with respect... 

A trial variation function

upper bound for the ground-state energy. One usually includes variational parameters in

variational integral W the function

ground-state wave function. 

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