Beryllium atom orbital energies

Alternative methods are based on the pioneering work of Hylleraas ([1928], [1964]). In these cases orbitals do not form the starting point, not even in zero order. Instead, the troublesome inter-electronic terms appear explicitly in the expression for the atomic wavefunction. However the Hylleraas methods become mathematically very cumbersome as the number of electrons in the atom increases, and they have not been very successfully applied in atoms beyond beryllium, which has only four electrons. Interestingly, one recent survey of ab initio calculations on the beryllium atom showed that the Hylleraas method in fact produced the closest agreement with the experimentally determined ground state atomic energy (Froese-Fischer [1977]). 

Figure 4 Energy changes during the formation of sp hybrid orbitals in the beryllium atom.

We describe in this Subsection the application of local-scaling transformations to the calculation of the energy for the lithium and beryllium atoms at the Hartree-Fock level [113]. (For other reformulations of the Hartree-Fock problem see [114] and referenres therein.) The procedure described here involves three parts. The first part is orbital transformation already discussed in Sect. 2.5. The second is intra-orbit optimization described in Sect. 4.3 and the third is inter-orbit optimization discussed in Sect. 4.6. 

Fig. 8. Intra- and inter-orbit energy convergence for an initial single-C wavefunction for the beryllium atom (energy in hartrees)...

The resulting functional has been evaluated for the initial orbitals of the lithium and beryllium atoms given in Section 4.6. Using the same process of intra- and inter-orbit optimization carried out in Section 4.6, but substituting, in each step, the value for /(r) given by the Fade approximant obtained from Eq. (189), energy values were obtained for the Li and Be atoms that are indistingishable from those previously calculated with the exact values of/(r), namely, when Eq. (40) is solved. 

The band of molecular orbitals formed by the 2s orbitals of the lithium atoms, described above, is half filled by the available electrons. Metallic beryllium, with twice the number of electrons, might be expected to have a full 2s band . If that were so the material would not exist, since the anti-bonding half of the band would be fully occupied. Metallic beryllium exists because the band of MOs produced from the 2p atomic orbitals overlaps (in terms of energy) the 2s band. This makes possible the partial filling of both the 2s and the 2p bands, giving metallic beryllium a greater cohesiveness and a higher electrical conductivity than lithium. 

A CASSCF(2,2)/6-31G calculation was done on one beryllium atom, using a simplified version of the procedure in Chapter 8 for molecules an orbital localization step is pointless for an atom, and in the energy calculation optimization is meaningless. First an STO-3G wavefunction was obtained and the atomic orbitals (AOs) were visualized this showed MOl, 2, 3, 4, and 5 to be, respectively, Is, 2s (both occupied), and three energetically degenerate unoccupied 2p orbitals. The active space was chosen to consist of the 2s and a 2p orbital, and a single-point (no optimization requested) CASSCF(2,2)/6-31G calculation was done. The energy was —14.5854725 hartree. 

The critical choice was made of a CASSCF(4,4)/6-31G calculation the active space is thus the degenerate filled 2s + 2s and 2s 2s pair of MOs, and the degenerate empty 2px + 2px and 2px — 2px pair of MOs. CASSCF(4,4) was chosen because it corresponds to the CASSCF(2,2) calculation on one beryllium atom in the sense that we are doubling up the number of electrons and orbitals in our noninteracting system. This calculation gave an energy of 29.1709451 hartree. We can compare this with twice the energy of one beryllium atom, 2 x — 14.5854725 hartree = —29.1709450 hartree. 

The 2p orbitals still exist for a beryllium atom they are simply empty. The atom could promote one electron to a 2p orbital, at the cost of some added energy. If this allowed it to contribute electron density to make two separate bonds, the added stability of the two bonds might be sufficient to justify the promotion energy. 

Each of the two sp-hybrid orbitals contains a single unpaired electron which is readily shared with a similar electron on another atom such as hydrogen. Notice how the sp hybrid orbital is intermediate in energy between the 2s and 2p orbitals of atomic beryllium. This means that it doesn t cost the beryllium atom much in energy to get its two outer electrons into the sp hybrid- what the beryllium atom must pay in order to promote its 2s electron up to the level of the sp hybrid, the atom more than gains in the energy lost when the previously promoted 2p electron now drops down to the sp-hybrid level. 

How are we to account for its combining with two chlorine atoms Bond formation is an energy-releasing (stabilizing) process, and the tendency is to form bonds—and as many as possible—even if this results in bond orbitals that bear little resemblance to the atomic orbitals we have talked about. If our method of mental molecule-building is to be applied here, it must be modifled. We must invent an imaginary kind of beryllium atom, one that is about to become bonded to two chlorine atoms. 

The main dijfference between a linear H3 system and beryllium hydride, BeH2, which is a linear molecule, is that the central atom now contributes two valence electrons and four atomic orbitals the 2s orbital, which is doubly occupied in an isolated ground state Be atom (configuration Is, 2s ), and the three 2p orbitals which are empty in Be but lie not too far above 2s in energy. Graphically, we have ... 

In the last 20 years, Cl calculations based on a single reference function have lost favor among practitioners. The principal shortcoming of these approaches is that they do not satisfy the property of size-consistency, which means that the Cl energy does not scale properly with the size of the system [112]. It is fairly easy to see why this is so. Consider two beryllium atoms, separated by a distance sufficiently large that the true physical interaction between the atoms vanishes. In a CISD description of this system, contributions to the wave function are excluded in which two electrons on each beryllium atom are in virtual orbitals, since these correspond to quadruply excited determinants and would require a method such as CISDQ or CISDTQ for their inclusion. However, the CISD wave function for a single beryllium atom contains all determinants with two electrons in virtual orbitals. Since Cl methods involve... 

El.30 Frontier orbitals of Be Recall from Section 1.9(c) Electron affinity, that the frontier orbitals are the highest occupied and the lowest unoccupied orbitals of a chemical species (atom, molecule, or ion). Since the ground-state electron configuration of a beryllium atom is ls 2s , the frontier orbitals are the 2s orbital (highest occupied) and three 2p orbitals (lowest unoccupied). Note that there can be more than two frontier orbitals if either the highest occupied and/or lowest unoccupied energy levels are degenerate. In the case of beryllium we have four frontier orbitals (one 2s and three 2p). 

As the 2s orbitals combine and spread out to form a band, it will overlap with the band generated by the combination of 2p orbitals. So, in solid beryllium (and similarly in other solid elements in Groups I—III (1,2 and 13)), the band generated by combination of atomic orbitals is not a pure 2s band, but one continuous band made by combination of both 2s and 2p orbitals. In consequence, there are unoccupied levels available in the energy band of solid beryllium into which electrons may be excited. This allows the development of metallic properties. 

This is why we took the beryllium atom and not just the helium atom, in which the energy difference between the orbital levels Is and 2s is much larger (i.e., the correlation energy much smaller). 

Because the major lobes of the hybrid orbitals are much more directed than the 2s orbital, a much stronger bond can be formed, and the extra energy released is more than sufficient to compensate for the energy required to promote an electron from the 2s to the 2p atomic orbital of beryllium. From the orientation of the two hybrid orbitals, the molecule is expected to be linear with a H-Be-H bond angle of 180°. 

An impressive success of the lEPA is its prediction of the ground state energy of the beryllium atom. The spin-orbital pair correlation energies obtained by... 

Figure 5.3 illustrates a beryllium atom with its energy levels. The atom is composed of 4 protons and 5 neutrons in the nucleus, and 4 electrons arranged in 2 shells (or orbital layers) outside the nucleus. The first shell contains 2 electrons and the second shell contains 2 electrons. 

The increase to 900 kJ mol" in the case of the beryllium atom is due to the increase in effective nuclear charge, offset by interelectronic repulsion of the two 2s electrons. The electron most easily removed is one of the pair in the 2s orbital. In the case of the boron atom, in spite of an increase in nuclear charge there is a decrease in the first ionization energy to 799 kJ mol. This is because the electron removed is from a 2p orbital, which is higher in energy than the 2s level. 

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