Helium ground states

Figure 4-1 Convergence of the correlation energy for the helium ground state with the number N of pair natural orbitals.

Recasting of correlated wavefunctions in helium (ground state)... 

As implied by the name, a correlated wavefunction takes into account at least some essential parts of the correlated motion between the electrons which results from their mutual Coulomb interaction. As analysed in Section 1.1.2 for the simplest correlated wavefunction, the helium ground-state function, this correlation imposes a certain spatial structure on the correlated function. In the discussion given there, two correlated functions were selected a three-parameter Hylleraas function, and a simple Cl function. In this section, these two functions will be represented in slightly different forms in order to make their similarities and differences more transparent. 

The result of such a calculation for Pls(r) of helium and how it compares with the hydrogenic case are shown in Fig. 7.6, and the differences can be seen clearly. For the total energy of the helium ground state the HF approach yields Eg = — 2.862 au. This is rather close to the experimental value, Eg(exp.) =... 

Abstract. With an eye on the high accuracy ( 10 MHz) evaluation of the ionization energy from the helium atom ground state, a complete set of order ma6 operators is built. This set is gauge and regularization scheme independent and can be used for an immediate calculation with a wave function of the helium ground state. 

Fig. 11.6. The 1200 eV noncoplanar-symmetric momentum profiles for the ground-state (n = 1) and summed n = 2 transitions in helium (Cook et al., 1984). Curves indicated DWIA, distorted-wave impulse approximation PWIA, plane-wave impulse approximation. The curves are calculated using a converged configuration-interaction expansion (McCarthy and Mitroy, 1986) for the helium ground state. The long-dashed curve is the distorted-wave impulse approximation for the Hartree—Fock ground state. Experimental data are normalised to the Is curve at low momentum. From McCarthy and Weigold (1991).

Figure J Helium ground state energy computed by two different methods. The Rivelino and Vianna calculations [108] vs the results of reference [117] considered here as the exact ones.

Figure 4 Helium ground state 11 S and the lowest triplet energy l3 S. The energy splitting between these two states grows as R diminishes.

Figure 5 Helium ground state energy as a function of distance D between the nucleus and the spherical box centre obtained by means of two basis sets four s orbitals for the first, and four s plus one p(z) for the second.

For example the helium ground state would have ct=lSj 2i/2 i i/2-i/2 Thus I q) is a normalized and antisymmetrized state with two freely propagating electrons, no positrons and no photons. The lowest order term, (84), becomes... 

To illustrate all-orders methods in a simple case, let us consider helium. We describe the helium ground-state in lowest order by the wave function... 

Now, let us consider the helium ground state. As a first step, we solve the Hartree-Fock equation... 

To go further, we turn to so-called all-order methods. We start by considering relativistic calculations of type carried out by Lindroth [38] and Blundell et al. [39], which account for all possible single and double excitations of the ground-state helium HF wave function. These calculations have the potential of giving an exact helium ground-state energy. 

Now, because of the opposed spins of the electrons in the helium ground state, the integrations over the spins reduce the first two integrals in equation 5.32 to integrals over the spatial coordinates of the electrons, while the same integrations over the spins reduces the third and fourth integrals in equation 5.32 to zero. Thus, equation 5.32 reduces to... 

The Hall-Roothaan equations for the case of the helium ground state appear, when equation 5.39 is written out in an appropriate linear combination of functions upon which the variation principle procedure can be applied to return the best energy in a calculation. For example, for the double-zeta basis of Slater functions used in the previous section, we have... 

Exercise 5.4. Formation of a 4-31) split-basis set for helium using the data of Table 5.1 and the calculation of the energy of the helium ground state. 

Therefore, for all finite systems, the asymptotic behavior of v (r) arises from the Fermi hole charge distribution p r, r ) and is given exactly by the structure of WP(r). The above analysis and conclusions are borne out as shown in the example of the Helium ground-state discussed in Sect. 5.2.1. 

Structure of the Exchange, Correlation and Correlation-Kinetic-Energy Fields and Potentials for the Helium Ground State... 

Recalling that equals 13.604 eV when the He reduced mass is used and putting Z = 2, we find for the first-order perturbation energy correction for the helium ground state ... 

Putting Z = 2, we get as our approximation to the helium ground-state energy - 21/ 6fe /ao = -(729/256)2(13. 4 eV) = -77.48 eV, as compared with the true value of -79.01 eV. Use of i instead of Z has reduced the error from 5.3% to 1.9%. In accord with the variation theorem, the true ground-state energy is less than the variational integral. 

The Bohr model gave the correct energies for the hydrogen atom but failed when applied to helium. Hence, in the early days of quantum mechanics, it was important to show that the new theory could give an accurate treatment of helium. The pioneering work on the helium ground state was done by Hylleraas in the years 192 1930. To allow for the effect of one electron on the motion of the other, Hylleraas used variational functions that contained the interelectronic distance ri2- One function he used is... 

The zeroth-order perturbation wave function (10.40) uses the full nuclear charge (Z = 3) for both the Is and 2s orbitals of lithium. We expect that the 2y electron, which is partially shielded from the nucleus by the two Is electrons, will see an effective nuclear charge that is much less than 3. Even the Is electrons partially shield each other (recall the treatment of the helium ground state). This reasoning suggests the introduction of two variational parameters and 2 into (10.40). 

EXAMPLE The He SCF calculation in Section 13.16 used a basis set of two STOs x and Xi-For the helium ground state treated with this basis set, (a) write down the configuration state functions (CSFs) that are present in the wave function in a full Cl treatment, and (b) carry out a Cl calculation that includes only doubly excited CSFs. 

For a configuration of closed subshells (for example, the helium ground state), we cm write only a single Slater determinant. This determinant is an eigenfunction of I and and is the correct zeroth-order function for the nondegenerate S term. A configuration... 

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