Helium atom: wavefunction

Table 3 Improvement of correlated helium atom wavefunctions by local-scaling transformations. The functional [p, g] is defined in Eq. (1). Energies in hartrees...

To see how this principle operates we need to include spin in the helium atom wavefunction by formally writing a for spin-up and P for spin-down, and indicating the electron with a particular spin by a number in parentheses. At first sight, there appear to be four spin possibilities for the two electrons in the helium atom ... 

This way, the Hamiltonian resembles two separate one-electron Hamiltonians added together. This suggests that perhaps the helium atom wavefunction is simply a combination of two hydrogen-like wavefunctions. Perhaps a sort of... 

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

The final wavefunction stUl contains a large proportion of the Is orbital on the helium atom, but less than was obtained without the two-electron integrals. 

In order to exemplify these ideas, we construct three initial or generating wavefunctions for the S state of the helium atom, and generate two types of... 

Note that in the present case the matrix elements depend on the final density p . Moreover, because this density is obtained from the transformed wavefunction, they also depend on the expansion coefficients. For this reason, Eq. (177) must be solved iteratively. Such a procedure has been applied - in a sample calculation - to the 2 S excited state of the helium atom. The upper-bound character of the energy corresponding to the energy functional for the transformed wavefunction [ p( r,- ) with respect to the exact energy is guaranteed by... 

Calculations of IIq(O) are very sensitive to the basis set. The venerable Clementi-Roetti wavefunctions [234], often considered to be of Hartree-Fock quality, get the sign of IIq(O) wrong for the sihcon atom. Purely numerical, basis-set-free, calculations [232,235] have been performed to establish Hartree-Fock limits for the MacLaurin expansion coefficients of IIo(p). The effects of electron correlation on IIo(O), and in a few cases IIq(O), have been examined for the helium atom [236], the hydride anion [236], the isoelectronic series of the lithium [237], beryllium [238], and neon [239] atoms, the second-period atoms from boron to fluorine [127], the atoms from helium to neon [240], and the neon and argon atoms [241]. Electron correlation has only moderate effects on IIo(O). 

In order to demonstrate the method, the simplest case, the ground state of the helium atom, will be used. Since the two-electron wavefunction is given by % = (lsO+, IsO-, one has to find the optimized orbitals Ru(r) which are part of starting point is the energy eigenvalue Eg... 

Table 2.2 2 Summary of four trial wavefunctions of helium atom in its ground state...

It is found that electronic wavefunctions must be antisymmetric with respect to the exchange of the coordinates of any two electrons. Hence, only the function given by expression (2.2.34) is acceptable for the helium atom ... 

For the excited configuration <7 aj 1, there are two states, one singlet and one triplet. This situation is similar to that found in the excited configuration ls12s1 of the helium atom. The singlet and triplet wavefunctions for these excited configurations are ... 

The interaction between a helium atom and the LiH molecule has been described using a SCVB wavefunction built up using just 25 structures. Interaction energies, computed along different approaches of the two moieties, compare extremely well with a corresponding traditional SCVB calculation using many more structures. Even a very small energy minimum of about 0.01 mHartree is perfectly reproduced for He at a distance of 7 =11 bohr from the centre of mass of the LiH molecule (collinear approach of He to H—Li). 

The study of the interactions among closed shell systems (van der Waals forces) represents a benchmark for theories of electron correlation. We report here the results of our variational MO-VB study of the interaction between two helium atoms [70]. Up to n=10 optimal virtuals are calculated and employed to generate MO-VB final wavefunctions of higher and higher accuracy, in the usual MO-VB form ... 

The simpler wavefunctions for helium atom, for example (8.5), can be interpreted as representing two electrons in hydrogenlike s orbitals, designated as a configuration. Pauli s exclusion principle, which states that no two electrons in an atom can have the same set of four quantum numbers, requires the two s electrons to have different spins one spin-up ora, the other spin-down or A product of an orbital with a spin function is called a spinorbital. For example, electron 1 might occupy a spinorbital which we designate... 

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