Multiplet States
****************

Calculations with the ADF engine yield results for one-determinant
electronic states, which are not always the 'true' states of the
molecule. The evaluation of the correct multiplet energies is not
trivial in this approach, see the section on multiplet energies
further below. The point is to evaluate a specific multiplet state as
a linear combination of selected one-determinant functions, each
computed in the field of the so-called Average-of-Configuration (AOC).
Typically, in an open shell system, the AOC is the spin-restricted
system in which all orbitals in the open shell are degenerate and
equally occupied. The AOC serves then as a fragment for the subsequent
calculations, in which the different open shell orbitals are occupied
differently by specifying the appropriate occupation numbers as
explained below.

**Important:** in these follow-up calculations it is imperative that
the results are obtained in the AOC field: no SCF convergence must be
carried out, because we only want to assign the electrons differently,
while keeping exactly the AOC orbitals. To achieve this, the follow-up
calculations must use the keyword SCF, and the iterations subkey must
be set to 0.

Since the ADF engine requires that the point-group symmetry matches
not only the nuclear frame but also to the electronic charge density
and MO occupations, these calculations must run in a lower point group
symmetry. Often you will also want to run the modified calculations
spin-*un*restricted. For an example, see the set of sample runs that
come with the package and the discussion in the Examples.

The calculation of the one-determinant states based on the AOC
reference state is controlled with the block "SLATERDETERMINANTS".

   SLATERDETERMINANTS file
   End

where the argument must be a file (including the path). The file must
be an ASCII file containing data in the same format as you would
supply in the data block, see below. All information on the file until
the *eof* must be suitable for the data block, but no record 'end' in
the file must be specified: only the *contents* of the data block.

Alternatively, you can use the SLATERDETERMINANTS block as follows

   SLATERDETERMINANTS
    title1
       irrep occups
       irrep occups
       ...
    subend
    title2
       irrep occups
       ...
    subend
    title3
       ...
    subend
    ...
   end

Each 'title' functions as a subkey, but is otherwise an arbitrary
string to label the resulting one-determinant calculation. Each such
subkey block contains the occupation numbers for a single one-
determinant calculation. It is necessary that the calculation uses the
reference AOC run as its only fragment file. The occupations in the
subkey blocks must be rearrangements of the AOC open-shell electrons.
In the Slaterdeterminants calculation you must explicitly specify the
point group symmetry in which you want to run; this must be a lower
symmetry than the AOC one, otherwise you couldn't rearrange the open
shell electrons. See the sections below on multiplet energies. A
sample run is included in Examples.

Each 'irrep occups' record specifies the occupations for the indicated
irrep in the usual way (see for instance the occupations key). The
irrep labels must correspond to the (lower) point group symmetry used
in the slaterdeterminants calculation. Note that in an unrestricted
calculation, occupation numbers must be given for both spins, using
the double slash (//) to separate the occupations for spin-\alpha and
spin-\beta.

In this setup, the program will, for each of the subkey blocks under
the slaterdeterminants key, execute an SCF calculation with only one
cycle, i.e. no convergence, where the start-up field is the fragment
field, i.e. the AOC field. So all one-determinant states in this
calculation are evaluated in the AOC field. The resulting energies for
the distinctly computed one-determinant states can then be combined to
the desired multiplet values, corresponding to how the multiplet
states are combinations of the one-determinant states.


Multiplet energies
==================

The energies of atomic and molecular multiplet states that correspond
to a given electron configuration can be calculated approximately with
the method suggested in ref. [1]. There it is indicated that it would
not be justified to take an arbitrary configuration-state function
(CSF), defined in general as a linear combination of determinants that
has specific spin and space symmetry properties, and use the
corresponding alpha and beta spin densities in a DFT energy
expression. The same holds true for the densities corresponding to the
average-of-configuration (see section 'DFT energy of a one-
determinantal wave function', the 'procedure' notes).

Therefore, we restrict ourselves to just computing the DFT energies of
single-determinant wavefunctions. Usually (but not always) this is
sufficient information to obtain the multiplet energies. The
procedure, which is explained in [1], requires knowledge of the CSFs
belonging to a given configuration. This means that a multiplet state
with specific L, ML and S, MS values has to be written as a linear
combination of the determinant wavefunctions that belong to the given
configuration. In Ref. [2] Claude Daul exploited symmetry in order to
simplify the relation between the multiplet energies and single-
determinant energies.

It is often advantageous to search for CSFs that consist of one
determinant only, since the energy of this determinant should
correspond directly to the multiplet energy. Sometimes there is
redundancy in the information and there may even be some
inconsistency: two determinants may exist that both are CSFs belonging
to the same multiplet state but yield somewhat different energies. We
will illustrate this for the Carbon atom example treated below.

The discrepancies are a measure of 'error bars' associated with the
theoretical multiplet energies. As a matter of fact, there are certain
symmetry relations between the energies of the determinants of a
configuration, calculated as the expectation value of the determinant
for the full many-electron Hamiltonian. An example is the equal energy
for the determinants of a p^1 configuration, whether the electron is
placed in the p_0 (= p_z) orbital or in the p_{+1} (=(p_x + i p_y) /
\sqrt{2}) orbital. This equality is not obtained with present-day
density functionals, leaving an ambiguity ('error bar') in the
determination of the energy. A more complete treatment of the symmetry
relations between determinant energies is given in [2].

**Warning:** However, there have occasionally been found inconsistent
results. A comparison to the results obtained by the procedure
outlined in [1] may show significant differences and the latter seem
more accurate and consistent.

**DFT energy of a one-determinantal wave function**

The determinant corresponds to a well-defined \rho^\alpha and
\rho^\beta. Suppose we are dealing with a d^2 configuration and we
wish to know the energy of

   D_1 = | d_2^\alpha (1) d_1^\alpha (2) |

where d_m has the Y_{2m} angular part. This determinant is a CSF of
the ^3 F multiplet

   D_1 = | ^3 F ; M_L = 3 ; M_S = 1 \rangle

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

   Z_{lm}^C = \frac{1}{\sqrt{2}} (Y_l^{-m} + Y_l^{-m*}) =
   \frac{1}{\sqrt{2}} (Y_l^{-m} + (-1^m) Y_l^{m})

   Z_{lm}^S = \frac{i}{\sqrt{2}} (Y_l^{-m} + Y_l^{-m*}) =
   \frac{i}{\sqrt{2}} (Y_l^{-m} + (-1^m) Y_l^{m})

with back transformations:

   Y_l^m = & \frac{1}{\sqrt{2}} (-1)^m (Z_{lm}^C + i Z_{lm}^S) \\
   Y_l^{-m} = & \frac{1}{\sqrt{2}} (Z_{lm}^C - i Z_{lm}^S)

Here the superscripts c and s stand for the cosine and sine types,
respectively, of combinations of exp(-im \phi) and exp(im \phi). This
yields explicitly:

   d_{z^2} & = d_0    \\ d_{xz} & = 1/\sqrt{2} (d_{-1} - d_{1})   \\
   d_{yz} & = i/\sqrt{2} (d_{-1} + d_1)   \\ d_{x^2-y^2} & =
   1/\sqrt{2} (d_{-2} + d_2)   \\ d_{xy} & = i/\sqrt{2} (d_{-2} - d_2)
   \\ d_{0} & = d_{z^2}   \\ d_{1} & = -1/\sqrt{2} (d_{xz} + i d_{yz})
   \\ d_{-1} & = 1/\sqrt{2} (d_{xz} - i d_{yz})   \\ d_{2} & =
   1/\sqrt{2} (d_{x^2-y^2} + i d_{xy})   \\ d_{-2} & = 1/\sqrt{2}
   (d_{x^2-y^2} - i d_{xy})

For D_1 we obtain:

   \rho_\alpha & = |d_2|^2 + |d_1|^2 = 1/2 |d_{x^2-y^2}|^2 + 1/2
   |d_{xy}|^2 + 1/2 |d_{xz}|^2 + 1/2 |d_{yz}|^2 \\ \rho_\beta & = 0

The fractional occupations have to be used in order to generate the
densities ρα and ρβ and the corresponding density matrices \rho_\alpha
and \rho_\beta. The density matrices can be used to calculate the
energy of D1 (and ^3 F) with respect to the energy of the 'master
fragment', which should be the restricted atom with d^2 occupation.
Other determinants of this configuration can be treated similarly to
obtain more multiplet energies of the d^2 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.
The ADF engine supports an automatic procedure to do this, using the
input key SLATERDETERMINANTS, see the ADF User's Guide, the Examples,
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
   d^2  configuration, the AOC calculation can be done in symmetry
   atom with occupation 2 in the d irrep. In case of a t_2g ^5  e_g ^1
   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 adf.rkf 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 \rho^\alpha  and \rho^\beta , i.e. to specific spin-
unrestricted occupations of the AOC orbitals. These densities
\rho^\alpha  and \rho^\beta  correspond to a symmetry group that will
in general be a subgroup of the symmetry group of the molecule. For
instance, the occupation (p_+1 \alpha)^1  in the case of an atomic p^1
configuration corresponds to

   \rho^\alpha = 1/2 p_x^2 + 1/2 p_y^2

with D_∞h  symmetry.

To obtain the energy of the determinant wave function 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 \rho^\alpha  and \rho^\beta  result. Note that the appropriate
(lower) symmetry point group 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 D_10  = |0^+  0^- | of the p^2  configuration of Carbon,
with density r=p_z ^2 . 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 adf.rkf
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 adf.rkf 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 period atoms**

In one of the next sections tables are given for the energy lowering
in going from the converged spherically symmetric spin-restricted atom
(the 'master' fragment) to specific one-determinant wavefunctions with
the orbital occupations as specified. Note that the p_x  and p_y
populations are always equal; only their sum is given. In many cases
the determinant corresponds to a specific state, which is then given
in the last column. For each atom, the first calculation is for the
spherically symmetric spin-*un*restricted atom. These tables are now
obsolete, all information needed to obtain the atomic reference
energies, i.e. the ground state multiplet energy with respect to the
AOC, can be found in ref. [3].

Examples worked out for all first period atoms:

**H**: Configuration (1s)^1. Only one determinant: | 1s \alpha (1) |

**He**: Configuration (1s)^2 . Closed shell.

**Li**: Configuration (2s)^1 . Only one determinant: | 2s \alpha (1) |

**Be**: Configuration (2s)^2 . Closed shell.

**B**: Configuration (2p)^1 .
   Ground multiplet ^2 P.

      D_1 & = |p_1 \alpha (1) | = | ^2 P ; M_L =1 ; M_S = 1/2 \rangle
      \\ \rho^\alpha & = 1/2(p_x-ip_y)(p_x+ip_y) = 1/2(p_x^2+p_y^2)

   The occupation numbers for D_1  are

      p_x^\alpha & = p_y^\alpha = 1/2 \\ p_z^\alpha & = p_x^\beta =
      p_y^\beta = p_z^\beta = 0

   Another determinant belonging to ^2 P is D_2  = |..p_0 \alpha (1)|
   with occupations p_z^\alpha = 1 and all other p-occupations zero.
   This determinant is 0.04 eV lower in energy than D_1 for LDA, but
   0.15 eV for BP.

**C**: Configuration p^2 .
   Multiplet states are ^3 P, ^1 S and ^1 D.  We use this atom as an
   example of the general procedure. First write down all determinants
   belonging to p^2  and group them according to M_S  and M_L  (1^+
   \equiv p_1 \alpha, ...)

   +-----------------------------------+-----------------------------------+-----------------------------------+
   | **Determinant**                   | M_S                               | M_L                               |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_1  = |1^+  1^- |                | 0                                 | 2                                 |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_2  = |1^+  0^+ |                | 1                                 | 1                                 |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_3  = |1^+  0^- |                | 0                                 | 1                                 |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_4  = |1^+  -1^+ |               | 1                                 | 0                                 |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_5  = |1^+  -1^- |               | 0                                 | 0                                 |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_6  = |1^-  0^+ |                | 0                                 | 1                                 |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_7  = |1^-  0^- |                | -1                                | 1                                 |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_8  = |1^-  -1^+ |               | 0                                 | 0                                 |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_9  = |1^-  -1^- |               | -1                                | 0                                 |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_10  = |0^+  0^- |               | 0                                 | 0                                 |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_11  = |0^+  -1^+ |              | 1                                 | -1                                |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_12  = |0^+  -1^- |              | 0                                 | -1                                |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_13  = |0^-  -1^+ |              | 0                                 | -1                                |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_14  = |0^-  -1^- |              | -1                                | -1                                |
   +-----------------------------------+-----------------------------------+-----------------------------------+
   | D_15  = |-1^+  -1^- |             | 0                                 | -2                                |
   +-----------------------------------+-----------------------------------+-----------------------------------+

   [image]

   The presence of a determinant with M_L  = 2, M_S  = 0 and no M_L  =
   2, M_S   0 determinant indicates the presence of a ^1 D multiplet,
   and E(^1 D) = E(D_1 ). There is also a ^3 P, the determinant with
   M_S  = 1, M_L  = 1. We should have E(^3 P) = E(D_2 ) = E(D_4 ). The
   two determinants D_3  and D_6  in the M_S  = 0, M_L  = 1 box will
   mix, and the solutions of the 2 by 2 secular problem will be E(^1
   D) and E(^3 P). Since the sum of the eigenvalues is equal to the
   sum of the initial diagonal elements, we have E(^1 D) + E(^3 P) =
   E(D_3 ) + E(D_6 ). We should also have E(D_3 ) + E(D_6 ) = E(D_1 )
   + E(D_2 ). Such a relation provides a test on the consistency of
   the results.   Finally we have the M_S  = 0, M_L  = 0 block. The
   sum of the energies of D_5 , D_8  and D_10  should be the sum of
   the energies of ^1 S, ^3 P and ^1 D. Since E(^3 P) and E(^1 D) are
   already known, E(^1 S) can be calculated.

   In the appendix we first locate for C the spherical unrestricted
   atom. Next we have E(D_4 ), yielding E(^3 P) = -1.345 eV (LDA +
   Becke). Next E(D_2 ) = E(^3 P) = -1.189 (always LDA + Becke). The
   difference is substantial: ~ 0.15  Next we have E(D_3 ) = - 0.812.
   Since E(D_6 ) = E(D_3 ), because \rho^\alpha (D_6 ) = \rho^\beta
   (D_3 ) and \rho^\beta (D_6 ) = \rho^\alpha (D_3 ), we should have
   2E(D_3 ) = -1.624 = E(^1 D) + E(^3 P). Therefore E(^1 D) = -1.624 -
   (-1.345) = - 0.279 or E(^1 D) = - 1.624 - (-1.189) = - 0.435. These
   numbers can be checked against E(D_1 ) which also should be E(^1
   D): E(D_1 ) =+ 0.044. The discrepancy is large! Finally, ^1 S can
   be obtained: E(D_10 ) = + 0.319 (D_8 ) = E(D_1 ) = + 0.044 E(D_5 )
   = E(D_1 ) = + 0.044. So 0.407 = E(^1 S) + E(^3 P) +E(^1 D).
   Different results for E(^1 S) are obtained depending on the E(^3 P)
   and E(^1 D) we choose: e.g. E(^1 S) = 0.407 -(-1.345) - (- 0.279) =
   2.031 or E(^1 S) = 0.407 - (-1.189) - (0.044) = 1.552. Comparing to
   experiment we might calculate the excitation energies w.r.t. E(^3
   P):

   +---------------------------+---------------------------+---------------------------+---------------------------+
   |                           | **calculated**            | **experimental**          | **HF**                    |
   +---------------------------+---------------------------+---------------------------+---------------------------+
   | ^3 P → ^1 D:              | 1.066 to 1.389            | 1.26                      | 1.55                      |
   +---------------------------+---------------------------+---------------------------+---------------------------+
   | ^3 P → ^1 S:              | 2.741 to 3.376            | 2.684                     | 3.78                      |
   +---------------------------+---------------------------+---------------------------+---------------------------+

**N**: Configuration p^3 .
   Ground multiplet ^4 S, corresponds to the spherical unrestricted
   atom, energy -2.943 eV. Other multiplets: ^2 P, ^2 D. According to
   the printed output for configuration p^3  we have |^2 D;M_L =2;M_S
   =1/2 = |p_1 ^\alpha  p_1 ^\beta  p_0 ^\alpha | = D_2   \rho^\alpha
   =  1/2 p_x ^2  +  1/2 p_y ^2  +      p_z ^2   \rho^\beta  =  1/2
   p_x ^2  +  1/2 p_y ^2  E(D_2 ) = - 0.745 according to the table in
   the Appendix (LDA + Becke). The energy of D_1 , with \rho^\alpha  =
   p_x ^2  +  p_y ^2 , \rho^\beta  = p_z ^2 , is E(|1A 2B 3A|) =
   -1.9702. The energy of D_3 , with \rho^\alpha  =  1/2 p_x ^2  +
   1/2 p_y ^2  +      p_z ^2 , \rho^\beta  = p_z ^2  corresponding to
   |1A 2A 2B| or |2A 2B 3A|, is E(D_3 ) = - 0.158. Finally, D_4  has
   \rho^\alpha  =  p_x ^2  +  p_y ^2 , \rho^\beta  =  1/2 p_x ^2  +
   1/2 p_y ^2 , corresponding to |1A 1B 3A| and |1A 3A 3B|, and E(D_4
   ) = - 0.109. The M_L = 1, M_S =1/2 determinants are |1A 1B 3A| and
   |1A 2A 2B|. Therefore E(^2 D) + E(^2 P) = E(D_4 ) + E(D_3 ), so
   E(^2 P) = - 0.109 - 0.158 - (- 0.745) = + 0.478. We can use D_1  in
   the M_L =0, M_S =1/2 block, from which we find E(^4 S) + E(^2 D) +
   E(^2 P) = 2E(D_2 ) =+ E(D_1 ). Hence E(^2 P) = -1.490 - 1.9702 - (-
   0.745) - (- 2.943) = + 0.2278.

**O**: Configuration p^4 .
   Multiplet states ^3 P, ^1 S, ^1 D. D_1 , with \rho^\alpha  =  p_x
   ^2  +  p_y ^2  +  p_z ^2 , \rho^\beta  =  p_z ^2  corresponds to
   |1A 2A 2B 3A| , the M_L =1, M_S =1 determinant of ^3 P: E(^3 P) =
   -1.836 D_2  with \rho^\alpha  =  p_x ^2  +  p_y ^2  +  p_z ^2 ,
   \rho^\beta  =  1/2 p_x ^2  +  1/2 p_y ^2 , corresponds to |1A 1B 2A
   3A|, the determinant of ^3 P: E(^3 P) = -1.568 D_3 , with
   \rho^\alpha  =  1/2 p_x ^2  +  1/2 p_y ^2  +  p_z ^2 , \rho^\beta
   =  p_x ^2  +  p_y ^2 , corresponds to |1A 1B 2A 3B|, and M_L =1,
   M_S =0 belonging to ^3 P as well as ^1 D.

**F**: Configuration p^5 .
   Ground multiplet ^2 P. As in B, we have two determinants with
   different energies belonging to this state: D_1  = |...(p_0 ^\alpha
   )^1  (p_0 ^\beta )^0 | → E(D_1 ) = - 0.715. D_2  = |(p_-1 )^2  (p_0
   )^2  (p_1 ^\alpha )^1  (p_1 ^\beta )^0 | → E(D_2 ) = - 0.467.

**Ne**: Configuration p^6 . Closed shell.

**Ground and Excited State Multiplets**

The computation of multiplets can be carried out with adf, using the
input block "SLATERDETERMINANTS".

The method described in [1] for the calculation of the energies of
proper spin and spatial symmetry adapted Configuration State Functions
is based on the calculation of the energies of single determinantal
wavefunctions. The densities corresponding to those Slater
determinants are inserted in the approximation used for the exchange-
correlation energy.

The procedure as detailed above is somewhat involved. Moreover, one
would like to have an easy procedure to calculate many determinants.
This is particularly desirable if one wishes to calculate the energies
of all Slater determinants of a given configuration, for instance if
one wishes to calculate certain averages in view of the
inconsistencies of the method.

We have implemented a semi-automatic procedure, using the block
"SLATERDETERMINANTS".

The general idea of this method is to first perform a restricted
calculation in the symmetry that is appropriate for the molecule. This
is called the average-of-configuration (AOC) calculation. This AOC
calculation generates the orbitals which will be used in all the
Slater determinants.

The AOC is the fragment that *must* be used in all subsequent
calculations. The subsequent calculations are characterized by having
the AOC as the (only) fragment, and by specifying the keyword
"SLATERDETERMINANTS". If an argument is given this must be a file
name. The named file should contain the occupations for the
determinants (see below). If no file name is specified, the
occupations should be specified in the data block.

The required information, on file or in the data block, is the
specification of the determinant or determinants that are to be
calculated in the form of orbital occupation numbers for the AOC
orbitals, using the irrep labels of the point group of the AOC
calculation, see below for format. All specified determinants will be
calculated, and the obtained energy will always be the energy
difference with respect to the AOC. Default occupations for all
subspecies of the AOC fragment are the occupations of the AOC itself.
Therefore, only the open (modified) subspecies have to be specified.

One has to be careful with respect to the point group symmetry to use
in the "SLATERDETERMINANTS" calculation. The density belonging to a
specific determinant is usually lower than the AOC symmetry (which is
the full symmetry group of the system). In that case this lower point
group symmetry has to be specified in the "SLATERDETERMINANTS"
calculation. Everything will always work fine if one just does not use
any symmetry at all (nosym). However, if for reasons of computational
efficiency one does want to use a subgroup of the system that
corresponds to the determinant density or densities, this is perfectly
possible. However, all the densities of the determinants specified
*must* then have this (or a higher) symmetry. The program does not
check on this, it is the user's responsibility to make sure that this
condition is satisfied for all the determinants. The only check that
is performed is that occupations of equivalent representations
(subspecies of one irrep) in the lower point group of the
"SLATERDETERMINANTS" run, that result from the specified occupations
of the subspecies of the AOC symmetry, are equal.

**Format of the input.**

**Important**: in the SlaterDeterminants calculations it is imperative
that the results are obtained in the AOC field: no SCF convergence
must be carried out, because we only want to assign the electrons
differently, while keeping the AOC orbitals exactly as they are. To
achieve this, the calculations should use the keyword SCF, and the
iterations subkey has to be set to 0 in the SCF data block.

Since adf requires that the point-group symmetry conforms not only to
the nuclear frame but also to the electronic charge density and mo
occupations, these calculations must run in a lower point group
symmetry. Often you will also want to run the modified calculations
spin-unrestricted.

For an example, see the set of sample runs that come with the package
and the comments in the examples.

See: SLATERDETERMINANTS block. Each 'title' functions as a subkey, but
is otherwise an arbitrary string to label the resulting one-
determinant calculation. Each such subkey block contains the
occupation numbers for a single one-determinant calculation. It is
necessary that the calculation uses the reference AOC run as its only
fragment file. The occupations in the subkey blocks must be re-
arrangements of the AOC open-shell electrons. In the
"SLATERDETERMINANTS" calculation you must explicitly specify the point
group symmetry in which you want to run. The \rho^\alpha  and
\rho^\beta densities of all determinants to be calculated must have
this point group symmetry, or a higher symmetry.

Each 'irrep occups' record specifies the occupations for the indicated
irrep in the usual way (see for instance the occupations key). The
irrep labels must correspond to the AOC point group symmetry used in
the AOC calculation, *not the point group symmetry used in the
``SLATERDETERMINANTS`` calculation!*. Note that in an unrestricted
calculation, occupation numbers must be given for both spins, using
the double slash (//) to separate the occupations for spin-alpha and
spin-beta.

In this setup, the program will for each of the subkey blocks under
the "SLATERDETERMINANTS" key execute an SCF calculation with only one
cycle, i.e. no convergence, where the start-up field is the fragment
field, i.e. the AOC field. So all one-determinant states in this
calculation are evaluated in the AOC field. The resulting energies for
the distinctly computed one-determinant states can then be combined to
the desired multiplet values.

**Example: Carbon** p^2

   SlaterDeterminants
   C(p2) ALFA: s=1, px=py=2/3, pz=2/3; BETA: s=1, p=0 ! title
      S 1 // 1 ! irrep name and occupations
      P:x 0.666666666666666666 // 0 ! another irrep, et cetera
      P:y 0.666666666666666666 // 0
      P:z 0.666666666666666666 // 0
      D:z2 0 // 0
      D:x2-y2 0 // 0
      D:xy 0 // 0
      D:xz 0 // 0
      D:yz 0 // 0
   SUBEND
   C(p2) ALFA: S=1, px=py=1, pz=0; BETA: s=1 !next (Sl.Determinant) title
      S 1 // 1
      P:x 1 // 0
      P:y 1 // 0
      P:z 0 // 0
      D:z2 0 // 0
      D:x2-y2 0 // 0
      D:xy 0 // 0
      D:xz 0 // 0
      D:yz 0 // 0
   SUBEND
   C(p2) ALFA: s=1, px=py=0.5, pz=1; BETA: s=1
      S 1 // 1
      P:x 0.5 // 0
      P:y 0.5 // 0
      P:z 1 // 0
      D:z2 0 // 0
      D:x2-y2 0 // 0
      D:xy 0 // 0
      D:xz 0 // 0
      D:yz 0 // 0
   SUBEND
   C(p2) ALFA: s=1, px=py=0.5, pz=0; BETA: s=1, px=py=0, pz=1
      S 1 // 1
      P:x 0.5 // 0
      P:y 0.5 // 0
      P:z 0 // 1
      D:z2 0 // 0
      D:x2-y2 0 // 0
      D:xy 0 // 0
      D:xz 0 // 0
      D:yz 0 // 0
   SUBEND
   C(p2) ALFA: s=1, px=py=0.5, pz=0; BETA: s=1, px=py=0.5, pz=0
      S 1 // 1
      P:x 0.5 // 0.5
      P:y 0.5 // 0.5
      P:z 0 // 0
      D:z2 0 // 0
      D:x2-y2 0 // 0
      D:xy 0 // 0
      D:xz 0 // 0
      D:yz 0 // 0
   SUBEND
   C(p2) ALFA: s= 1, px=py=0, pz=1; BETA: s=1, px=py=0, pz=1
      S 1 // 1
      P:x 0 // 0
      P:y 0 // 0
      P:z 1 // 1
      D:z2 0 // 0
      D:x2-y2 0 // 0
      D:xy 0 // 0
      D:xz 0 // 0
      D:yz 0 // 0
   SUBEND

In the example the AOC calculation is the carbon atom in spherical
symmetry (symmetry name atom).

Several spin states can be generated from this AOC set of orbitals,
but they all have a lower symmetry than the AOC. In the example the
point group D_∞h  (DLIN) could be used in the "SLATERDETERMINANTS"
calculation. In D_∞h  the *p* orbitals split into two sets, *p**_x *
and *p**_y * occur in \pi_x  and \pi_y  respectively, so their
occupations must be identical, and *p**_z * is a \Sigma_u  orbital.

In the block "SLATERDETERMINANTS" (or in the file) we now specify the
occupations for the subspecies of the atom irreps of a specific Slater
determinant and the program will sort out the corresponding
occupations in the d(lin) symmetry.

In all cases the orbitals used for the energy calculation(s) will be
the self-consistent AOC orbitals.

In the given example, the first set of occupations does not correspond
to a Slater determinant, but is the spin-polarized spherical case with
the *p* electrons evenly distributed over all components.

**LDA results, with and without GGA (Becke-Perdew)**

Energy changes (eV) for atoms going from restricted to (one-cycle)
unrestricted. Results between parentheses are for *converged*
unrestricted calculations.

All calculations have been performed in D_∞h  symmetry, since p_x  and
p_y  always had equal occupations and therefore could occur as \pi_u
-x and \pi_u -y partners of the \Pi_u  irrep.

+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
| El.   | Occupations                                    | LDA               | LDA+Becke         | BP                |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | alpha-spin             | beta-spin             |                                                           |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | s    | px+py   | pz    | s   | px+py   | pz    |                                                           |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
| H     | 1    | 0       | 0     | 0   | 0       | 0     | -0.868 (-0.898)   | -0.758 (-0.837)   | -0.889 (-0.948)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
| Li    | 1    | 0       | 0     | 0   | 0       | 0     | -0.231 (-0.235)   | -0.195 (-0.207)   | -0.249 (-0.256)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
| Be    | 1    | 0       | 0     | 1   | 0       | 0     | 0.000  (0.000)    | 0.000 (0.000)     | 0.000 (0.000)     |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
| B     | 1    | 2/3     | 1/3   | 1   | 0       | 0     | -0.247 (-0.255)   | -0.231 (-0.242)   | -0.276 (-0.281)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 0       | 1     | 1   | 0       | 0     | -0.295 (-0.321)   | -0.436 (-0.474)   | -0.448 (-0.485)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 1       | 0     | 1   | 0       | 0     | -0.266 (-0.279)   | -0.296 (-0.316)   | -0.333 (-0.348)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
| C     | 1    | 4/3     | 2/3   | 1   | 0       | 0     | -1.163 (-1.203)   | -1.109 (-1.158)   | -1.252 (-1.285)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 2       | 0     | 1   | 0       | 0     | -1.152 (-1.211)   | -1.271 (-1.345)   | -1.372 (-1.436)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 1       | 1     | 1   | 0       | 0     | -1.152 (-1.197)   | -1.134 (-1.189)   | -1.267 (-1.307)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 1       | 0     | 1   | 0       | 1     | -0.462 (-0.506)   | -0.726 (-0.812)   | -0.778 (-0.868)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 1       | 0     | 1   | 1       | 0     | 0.159  (0.150)    | 0.039 (0.044)     | 0.087 (0.086)     |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 0       | 1     | 1   | 0       | 1     | 0.730  (0.668)    | 0.322 (0.319)     | 0.480 (0.450)     |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
| N     | 1    | 2       | 1     | 1   | 0       | 0     | -2.936 (-3.032)   | -2.827 (-2.943)   | -3.101 (-3.190)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 2       | 0     | 1   | 0       | 1     | -1.362 (-1.454)   | -1.811 (-1.972)   | -1.943 (-2.108)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 1       | 1     | 1   | 1       | 0     | -0.581 (-0.618)   | -0.688 (-0.745)   | -0.746 (-0.801)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 1       | 1     | 1   | 0       | 1     | 0.178  (0.088)    | -0.104 (-0.158)   | -0.069 (-0.140)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 2       | 0     | 1   | 1       | 0     | 0.197  (0.135)    | -0.077 (-0.109)   | -0.011 (-0.053)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
| O     | 1    | 2       | 1     | 1   | 2/3     | 1/3   | -1.400 (-1.477)   | -1.361 (-1.447)   | -1.480 (-1.552)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 2       | 1     | 1   | 0       | 1     | -1.442 (-1.583)   | -1.698 (-1.836)   | -1.816 (-1.957)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 2       | 1     | 1   | 1       | 0     | -1.422 (-1.515)   | -1.470 (-1.568)   | -1.590 (-1.678)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 1       | 1     | 1   | 2       | 0     | -0.564 (-0.623)   | -0.866 (-0.960)   | -0.913 (-1.013)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 1       | 1     | 1   | 1       | 1     | 0.358  (0.321)    | 0.255 (0.237)     | 0.292 (0.266)     |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 2       | 0     | 1   | 2       | 0     | 1.323  (1.220)    | 0.825 (0.789)     | 0.992 (0.932)     |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
| F     | 1    | 2       | 1     | 1   | 4/3     | 2/3   | -0.374 (-0.398)   | -0.366 (-0.391)   | -0.394 (-0.416)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 2       | 1     | 1   | 2       | 0     | -0.323 (-0.409)   | -0.605 (-0.686)   | -0.627 (-0.715)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+
|       | 1    | 2       | 1     | 1   | 1       | 1     | -0.349 (-0.389)   | -0.401 (-0.441)   | -0.427 (-0.467)   |
+-------+------+---------+-------+-----+---------+-------+-------------------+-------------------+-------------------+

[1] T. Ziegler, A. Rauk and E.J. Baerends, *On the calculation of
    Multiplet Energies by the Hartree Fock Slater method*, Theoretica
    Chimica Acta 43, 261 (1977)

[2] C. Daul, *DFT applied to excited states*, International Journal of
    Quantum Chemistry 52, 867 (1994)

[3] E.J. Baerends, V. Branchadell and M. Sodupe, *Atomic reference-
    energies for density functional calculations*, Chemical Physics
    Letters 265, 481 (1997)
