Lowest energy principle

Assume now that two different external potentials (which may be from nuclei), Vext and Vgjjj, result in the same electron density, p. Two different potentials imply that the two Hamilton operators are different, H and H, and the corresponding lowest energy wave functions are different, and Taking as an approximate wave function for H and using the variational principle yields... 

The lowest-energy orbitals fill up fust, according to the order Is —> 2s —> 2p —> 3s —> 3p — 4s — 3d, a statement called the aufbciii principle. Note that the 4s orbital lies between the 3p and 3d orbitals in energy. 

Aufbau principle The principle that states that the lowest-energy orbitals fill first when electrons are added to successive elements in the periodic table. 

Nevertheless, the situation is not completely hopeless. There is a recipe for systematically approaching the wave function of the ground state P0> i- c., the state which delivers the lowest energy E0. This is the variational principle, which holds a very prominent place in all quantum-chemical applications. We recall from standard quantum mechanics that the expectation value of a particular observable represented by the appropriate operator O using any, possibly complex, wave function Etrial that is normalized according to equation (1-10) is given by... 

Now that we have decided on the form of the wave function the next step is to use the variational principle in order to find the best Slater determinant, i. e., that one particular Osd which yields the lowest energy. The only flexibility in a Slater determinant is provided by the spin orbitals. In the Hartree-Fock approach the spin orbitals (Xi 1 are now varied under the constraint that they remain orthonormal such that the energy obtained from the corresponding Slater determinant is minimal... 

At zero temperature the electrons in a solid occupy the lowest energy levels compatible with the Pauli exclusion principle. The highest energy level occupied at T = 0 is the Fermi level, Ep. For metals the Fermi level and the electrochemical potential are identical at T = 0, since any electron that is added to the system must occupy the Fermi level. At finite temperatures Ep and the electrochemical potential p of the electrons differ by terms of the order of (kT)2, which are typically... 

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