Helium atom correlation energy

For all intents and purposes then, we are concerned here with the CCSD (coupled cluster with all single and double substitutions [30]) correlation energy. Its convergence is excruciatingly slow Schwartz [31] showed as early as 1963 that the increments of successive angular momenta l to the second-order correlation energy of helium-like atoms converge as... 

With respect to correlation functionals, corrections to the correlation energy density following Eq. (8.29) include B88, P86, and PW91 (which uses a different expression than Eq. (8.27) for the LDA correlation energy density and contains no empirical parameters). Another popular GGA correlation functional, LYP, does not correct the LDA expression but instead computes the correlation energy in toto. It contains four empirical parameters fit to the helium atom. Of all of the correlation functionals discussed, it is the only one tliat provides an exact cancellation of the self-interaction error in one-electron systems. 

A quantitative analysis of the non-dynamical and dynamical components of the correlation energy for the helium atom is given in Table 2. 

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

FIGURE 6. Examples for comparison of molecular state data based on first and second order perturbation (a) Correlation of the vertical 7r-ionization energies of heterobenzenes C5H5X36 with atomic ionization energies of elements X allowing a correct prediction for silabenzene15 37 and (b) second order perturbation in silylacetylene as visualized by its (helium I) photoelectron spectrum... 

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,... 

The data points are fitted in a least-square sense to a fourth degree polynomial, and the properties thereby obtained are presented in Table 3. Since the atom possesses spherical symmetry there is only a single independent component of the a-tensor as well as the y-tensor. The curvature of the energy, or the polarizability, at the SCF level differs by less than 5% compared to the FCI result, and the MP2 value captures slightly more than half of the correlation effect. Electron correlation plays a more important role in the determination of the fourth-order property y. Again the MP2 method captures slightly more than half of the total contribution, which amounts to 21% at the FCI level of theory. The trends we have seen here in the example of the helium atom are more or less representative for closed-shell molecules in general. 

Total energy (Etotai) and full Cl correlation energy (Ecorr) for the beryllium atom and the helium dimer (Rt = 5.6 bohr) obtained with the numerical molecular basis sets. All energies are in Hartree. 

Calculation of the ground state energy of the helium atom is a critical case as well because it is the first example of correlation energy, the difference between the Hartree-Fock energy and the exact value. The energy required to remove one electron from a neutral He atom is the first ionization potential... 

In the late 60 s Gimarc [101] analyzed the confined helium atom problem by systematically studying the correlation energy in a two electron atom. The correlation energy is defined as the difference between the Hartree-Fock energy and the exact value,... 

The correlation energies for free (unconfined) H, He, Li+ and Be++ were well known at that time, and so Gimarc wanted to analyze, in particular, how the correlation energy changes as a function of the box radius for the confined helium atom isoelectronic series. Gimarc performed a number of variational calculations based on the following wave functions ... 

Table 2 Correlation energy (CE) estimated as the difference between the wave function expanded with 40 Hylleraas functions (H-WF) and the HF wave function obtained with optimized exponents, for the lowest singlet state of confined helium atom. All quantities are in hartrees...

As in the closed-shell case, we estimate the correlation energy involved in the lowest triplet state of the confined helium atom. In Table 6 we report the total energy obtained from Table IV of Ref. [19], also the HF results from this work, and results obtained from correlated methods. We see that the results obtained with the HF method, coupled with exponent optimization, compete with the other correlated methods, since for some confinement radii our results are below those reported by the other authors. Comparing our Hartree-Fock results with those obtained with 40-terms Hylleraas wave... 

Truncate this operator to third order with CCSD(T) and it still reproduces an estimated 97% of the correlation description [9]. (It is worth noting that methods exist which explicitly include the interelectronic potential. Recent calculations on the helium atom using Hylleraas-type r12 methods were able to match the exact non-relativistic energy to an astounding 10 12 kcal/mol [10].)... 

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