Energies of Isolated Atoms

The ΔH_f calculated by semiempirical methods is defined as the energy in kcal.mol^-1 required to form one mole of the system in the gas phase at 298K from its elements in their standard state:

In order to calculate ΔH_f , the quantity E_isol must be determined; this is the energy required to form the isolated atom from its valence electrons:

In the calculation of E_elect, the energy of valence electrons is defined as zero, likewise in calculating E_nuc, the energy of the isolated nucleus is defined as zero, therefore the calculation of E_isol simplifies to the calculation of E_{neutral atom}.

The energy of E_eisol is the energy released when the valence electrons are added to the nucleus. For example, for the hydrogen atom, this would be U_ss. For poly-electronic atoms, the electron-electron interactions must be included, in addition to the one-electron contributions. Most elements have open shell ground states, and for these systems, the nature of the state is important.

For all main group elements, that is, elements with valence shell configurations of the form ns^anp^b, other than the alkali metals, the value of E_isol is given by:

in which c=min(b(b-1)/2,(6-b).(5-b)/2). Except for the H_pp term, all the contributions to E_isol are obvious. Non-zero H_pp terms occur when there are two or more unpaired electrons in the ground state, in which case there is an exchange stabilization that is otherwise absent.

Because H_pp is usually written as 1/2(G_pp-G_p2), the expression for systems with 2 to 4 p electrons is recast as:

or

For the alkali metals, the equation for E_isol is the same as that for hydrogen.

For the transition metals, the coefficients for the d-d interactions are more complicated.

The general form for E_isol for a transition metal of configuration s^mdⁿs, in which there are m_a α s-electrons and m_b β s-electrons, and n_a α d-electrons and n_b β d-electrons, and the total angular quantum number is L, is:

As might be imagined, derivation of this expression is by no means obvious, particularly the terms for G_dd² and G_dd⁴. Interested readers are referred to Racah's paper in Phys Rev, 61, 186 (1942). In this, Racah derived an expression for the d orbital energy of the ground state in terms of three quantities, A, B, and L, the total angular momentum:

The quantities A and B, and a third quantity, C, not used here, are related to the G_k as follows:

Using Racah's equation, derivation of E_isol is straightforward. In texts on transition metal ion theory, the quantities G_dd⁰, G_dd⁰, and G_dd⁰ are usually represented by the symbols F₀, F₂, and F₄, respectively. However, care should be exercised when reading these texts: sometimes other quantities, F⁰, F², and F⁴ are used. The relationship between these three sets of symbols is as follows:

Because the coefficients for the two electron terms are so complicated, values for all elements likely to be parameterized for semiempirical methods are presented in the Table. From this table, the values of some coefficients are readily derived. Thus for the s-d coulomb integral, G_sd, the coefficient is simply the number of s electrons times the number of d electrons. One s-d exchange integral, H_sd, exists for each electron in the s shell for which there is an electron of the same spin in the d shell. For elements with two s electrons, this is simply the number of d electrons, for elements with one s electron, the Aufbau principle indicates that the d shell with higher occupancy has the same spin as that of the s electron. Finally, the coefficients for the simple d-d repulsion integral, G_dd^{0Note also that there are no elements with both p and d valence electrons, therefore terms of the type G_pd are not necessary.}