1.1 DFT energy of a one-determinantal wavefunction

The determinant corresponds to a well defined ρ^α and ρ^β. Suppose we are dealing with a d² configuration and we wish to know the energy of

D₁ = |d₂^α(1) d₁^α(2)|

where d_m has the Y_2m angular part. This determinant is a CSF of the ³F multiplet:

D₁ = |³F;M_L=3;M_S=1⟩

We can easily transform to the real spherical harmonics that are used in ADF:

Z_lm^c = 1/√2 (Y_l^-m + Y_l^-m*) = 1/√2 [Y_l^-m + (-1)^mY_l^m]

Z_lm^s = i/√2 (Y_l^-m + Y_l^-m*) = i/√2 [Y_l^-m + (-1)^mY_l^m]

with back transformations:

Y_l^m = 1/√2 (-1)^m [Z_lm^c + i Z_lm^s]

Y_l^-m = 1/√2 [Z_lm^c - i Z_lm^s]

Here the superscripts c and s stand for the cosine, respectively sine type of combinations of exp(-imφ) and exp(imφ). This yields explicitly:

d_z² = d₀
d_xz = 1/√2 (d_-1 - d₁)
d_yz = i/√2 (d_-1 + d₁)
d_x²-y² = 1/√2 (d_-2 + d₂)
d_xy = i/√2 (d_-2 - d₂)

d₀ = d_z²
d₁ = -1/√2 (d_xz + i d_yz)
d_-1 = 1/√2 (d_xz - i d_yz)
d₂ = 1/√2 (d_x²-y² + i d_xy)
d_-2 = 1/√2 (d_x²-y² - i d_xy)

For D₁ we obtain:

ρ^α = |d₂|² + |d₁|² = 1/2 |d_x²-y²|² + 1/2 |d_xy|² + 1/2 |d_xz|² + 1/2 |d_yz|²
ρ^β = 0

The fractional occupations have to be used in order to generate the densities ρ^α and ρ^β and the corresponding density matrices P^α and P^β. The density matrices can be used to calculate the energy of D₁ (and ³F) with respect to the energy of the 'master fragment', which should be the restricted atom with d² occupation. Other determinants of this configuration can be treated similarly to obtain more multiplet energies of the d² configuration.

Below is an example of how you can obtain determinant energies 'by hand', i.e. by carrying out a specific sequence of ADF calculations. ADF supports an automatic procedure to do this, using the input key SLATERDETERMINANTS, see the ADF User's Guide, the Examples document, and below.

Procedure

1 Determine a set of orbitals belonging to the given configuration. These orbitals are generated in what we call the average-of-configuration (AOC) calculation. This is a spin-restricted SCF calculation where the electrons of the configuration are distributed equally over the subspecies of the open shell irreps in order to retain the A1 symmetry of the total density in the symmetry group of the molecule. For instance, in case of an atomic d² configuration, the AOC calculation can be done in symmetry atom with occupation 2 in the d irrep. In case of an t_2g⁵ e_g¹ configuration of an octahedral complex, the AOC calculation requires an occupation of 5 electrons in the t_2g, and 1 electron in the e_g.
The result file TAPE21 of the AOC calculation has to be saved, to be used as a fragment file in the subsequent calculations.

2 The AOC is used as a fragment in all subsequent calculations that are performed to obtain single determinant energies. This means that those single determinant energies are always with respect to the AOC energy. This is a case where there is only one "fragment", which is actually the complete system, but in a different electronic configuration and in possibly a different symmetry group.

Suppose that a single determinant corresponds to spin-up and spin-down densities ρ^α and ρ^β, i.e. to specific spin-unrestricted occupations of the AOC orbitals. These densities ρ^α and ρ^β correspond to a symmetry group that will in general be a subgroup of the symmetry group of the molecule. For instance, the occupation (p₊₁α)¹ in the case of an atomic p¹ configuration corresponds to

ρ^α = 1/2 p_x² + 1/2 p_y²

with D_∞h symmetry.

To obtain the energy of the determinant wavefunction we must now perform one cycle (iterations= 0 in the block key SCF) of a spin-unrestricted calculation, with AOC as (the only) fragment with alpha and beta occupation numbers (using the input key occupations) such that ρ^α and ρ^β result. Note that the appropriate (lower) symmetry pointgroup must be specified in the input file.

Occasionally, the single determinant corresponds to a closed shell configuration in the appropriate lower symmetry, for instance determinant D₁₀ = |0⁺ 0^-| of the p² configuration of Carbon, with density r=p_z². In that case the one-cycle calculation can of course be spin-restricted.

N.B.1. One cycle will regenerate the SCF orbitals of AOC, if the same field is used as the converged AOC field. This will actually be the case because the starting potential is taken from the fragment TAPE21 file.
The key modifystartpotential must not be used (the density should be distributed equally over the spins).

N.B.2. After diagonalization in the one-cycle run, the AOC orbitals have been obtained again and are occupied as specified. The ('bonding') energy is calculated from the resulting charge density.

Remarks:

• If one does not perform just one cycle, but instead converges the unrestricted calculation,
  the energy will be lowered by 'polarization' of the orbitals.
  It is theoretically not so clear what the status of such converged energies is.
  Usually the energy lowering is in the order of 0.1 eV, but it may be quite a bit larger.
• It is not necessary to use AOC as fragment in the single-determinant runs.
  It is also perfectly allowed to run all calculations (ground state, AOC, determinants)
  from one set of fragments, for instance the standard atomic fragments.
  Since we must arrange that the one-cycle determinant calculations use the AOC field,
  so as to reproduce the AOC orbitals, we must then supply the result file TAPE21 of the AOC
  run as a restart file, using the key restart; see the adf User's Guide.
  Of course, in such an approach the computed energies are with respect to another reference,
  for instance the restricted atoms.

Results for first and second period atoms

85.10.201.17

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