Helium atom, calculations

The energy of the helium atom calculated above is the first-order energy, which differs from the true energy by an amount called the correlation energy this is a measure of the tendency of the electrons to avoid each other. The simplest improvement to the trial wave function is to allow Z in (6.29) to be a variable parameter, which we call (not to be confused with the spin-orbit coupling parameter in equation (6.20)) Z in the Hamiltonian (6.23) remains the same. The expression for the calculated energy,... 

Run the Herman-Skillman program for the helium atom calculation. Select hs.exe and pressing the return button, or simply double click on the file icon. The program runs in interactive mode. When asked for the input file, type he.in, press return and type c he.out when asked to name the output file. The program runs in DOS mode and returns to WINDOWS after execution is complete. ... 

Figure 5.4a The worksheet Dzcalc in fig5-4.xls for the calculation of the various terms in the Fock matrix for the helium atom calculation using the Slater double-zeta basis from Table 1.3 and in the second part of the figure, the outputs from each iteration until convergence is reached after nine cycles of calculation.

Table 1 Ground-state energies of the helium atom calculated with the g function given by Eq. (10) and the initial function xjro given by Eq. (11) ...

Alpha particles produced from radioactive decays eventually pick up electrons from the surroundings to form helium atoms. Calculate the volume (mL) of... 

The so called Hartree-Fock equations represent a pseudo-eigenvalue problem which requires an iterative approach and for which the use of computers is ideally suited. The total energy of the helium atom calculated in this way shows an error of approximately 1.5% as compared with the experimental value. 

A Self-Consistent Field Variational Calculation of IP for the Helium Atom... 

Use Mathcad to calculate the first approximation to the SCF energy of the helium atom... 

To make matters worse, the use of a uniform gas model for electron density does not enable one to carry out good calculations. Instead a density gradient must be introduced into the uniform electron gas distribution. The way in which this has been implemented has typically been in a semi-empirical manner by working backwards from the known results on a particular atom, usually the helium atom (Gill, 1998). It has thus been possible to obtain an approximate set of functions which often serve to give successful approximations in other atoms and molecules. As far as I know, there is no known way of yet calculating, in an ab initio manner, the required density gradient which must be introduced into the calculations. 

For atoms with more than two electrons, it is very difficult to obtain such a small absolute error in the energy as in the helium case, but, within an isoelectronic sequence, the relative error will, of course, go down rapidly with increasing atomic number Z. The method of superposition of configurations has been used successfully in a number of applications, particularly by Boys (1950-) and Jucys (1947-), and, for a more detailed survey of the work on atoms, we will refer to the special table on atomic calculations in the bibliography. This is a field of rapid development, where one can expect important new results within the next few years. 

Calculate the apparent volume (in pnt1) and radius (in pm) of a helium atom as determined from the van der Waals parameters,... 

As an example we may calculate the energy of the helium atom in its normal state (24). Neglecting the interaction of the two electrons, each electron is in a hydrogen-like orbit, represented by equation 6 the eigenfunction of the whole atom is then lt, (1) (2), where (1) and (2) signify the first and the second electron. 

Numerical values of E > and E + for the helium atom (Z = 2) are given in Table 9.1 along with the exact value. The unperturbed energy value E l has a 37.7% error when compared with the exact value. This large inaccuracy is expected because the perturbation H in equation (9.80) is not small. When the first-order perturbation correction is included, the calculated energy has a 5.3% error, which is still large. 

Nor were the astronauts the first to provide information about elements outside the earth. Astronomers, scanning the sky for light spectra from stars and galaxies, have calculated that the universe is composed of about 90% hydrogen (atomic number 1) and 9% helium (atomic number 2). All the other elements, which make up most of our earth, are an insignificant 1 percent of the total universe. 

We have just explained that the wave equation for the helium atom cannot be solved exacdy because of the term involving l/r12. If the repulsion between two electrons prevents a wave equation from being solved, it should be clear that when there are more than two electrons the situation is worse. If there are three electrons present (as in the lithium atom) there will be repulsion terms involving l/r12, l/r13, and l/r23. Although there are a number of types of calculations that can be performed (particularly the self-consistent field calculations), they will not be described here. Fortunately, for some situations, it is not necessary to have an exact wave function that is obtained from the exact solution of a wave equation. In many cases, an approximate wave function is sufficient. The most commonly used approximate wave functions for one electron are those given by J. C. Slater, and they are known as Slater wave functions or Slater-type orbitals (usually referred to as STO orbitals). 

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). 

The corrugation of the charge density on metal surfaces can be obtained from first-principles calculations or helium scattering experiments. The theory and the experiments match very well. A helium atom can reach to about 2.5-3 A from the top-layer nuclei. At that distance, the repulsive force between the helium atom and the surface is already strong. The corrugation at that distance is about 0.03 A, from both theory and experiments. For STM,... 

In order to appreciate the size of the basis sets required for fully converged calculations, consider the interaction of the simplest radical, a molecule in a electronic state, with He. The helium atom, being structureless, does not contribute any angular momentum states to the coupled channel basis. If the molecule is treated as a rigid rotor and the hyperfine structure of the molecule is ignored, the uncoupled basis for the collision problem is comprised of the direct products NMf ) SMg) lnii), where N = is the quantum number... 

Pecul calculated the 1J(3He,3He) coupling in ,154 i.e., one of the weakest bonded van der Waals complexes, using full Cl46 and EOM-CCSD methods.44,155 158 She found that in this complex the FC term of such coupling decreases exponentially with the d(He-He) distance being He, 3He) 22 Hz, for d(He He) = 4 au and falling below 0.1 Hz for d(He-He) = 7 au. Pecul concluded that the main FC coupling pathway is the overlap between the electronic clouds of both helium atoms and that its efficiency does not depend on whether this corresponds to an attractive or repulsive interaction. Similarly, Pecul et al.159 carried out calculations based on... 

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