6.4 Restricted Open-Shell Hartree–Fock

6.4.1 Brief Description

The spin-restricted open-shell Hartree–Fock method (ROHF) can always be chosen to systems where all unpaired spins are parallel. The TURBOMOLE keywords for such a case (one open shell, triplet e\(_g^2\,\)) are:

$open shells  type=1
  eg     1               (1)
$roothaan   1
  a=1  b=2

It can also treat more complicated open-shell cases, as indicated in the tables below. In particular, it is possible to calculate the [xy]\(^{\mbox{\rm singlet}}\) case. As a guide for expert users, complete ROHF TURBOMOLE input for O\(_2\) for various CSFs (configuration state function) is given in Section 26.6. Further examples are collected below.

The ROHF ansatz for the energy expectation value has a term for interactions of closed-shells with closed-shells (indices \(k,l\)), a term for purely open-shell interactions (indices \(m,n\)) and a coupling term (\(k,m\)): \[\begin{align*} E =& 2 \sum_{k}h_{kk} + \sum_{k,l}(2J_{kl}-K_{kl}) \\ & + f[ 2\sum_{m}h_{mm} + f\sum_{m,n}(2aJ_{mn}-bK_{mn}) + 2\sum_{k,m}(2J_{km}-K_{km}) ] \end{align*}\] where \(f\) is the (fractional) occupation number of the open-shell part (\(0<f<1\)), and \(a\) and \(b\) are the Roothaan parameters, numerical constants which depend on the particular configuration of interest.

6.4.2 One Open Shell

Given are term symbols (up to indices depending on actual case and group) and \(a\) and \(b\) coefficients. \(n\) is the number of electrons in an irrep with degeneracy \(n_{ir}\). Note that not all cases are Roothaan cases.

All single electron cases are described by:

\(a = b = 0\)

Example

The \(4d^9\,5s^2\) \(^2\)D state of Ag, in symmetry \(I\)

$closed shells
  a       1-5                                    ( 2 )
  t1      1-3                                    ( 2 )
  h       1                                      ( 2 )
$open shells type=1
  h       2                                      ( 9/5 )
$roothaan         1
  a = 80/81      b = 80/81

6.4.3 More Than One Open Shell

A Half-filled shell and all spins parallel

All open shells are collected in a single open shell and \[\begin{equation*} a = 1 \qquad b = 2 \end{equation*}\]

Example:

The \(4d^5\,5s^1\) \(^7\)S state of Mo, treated in symmetry I

$roothaan         1
  a = 1      b = 2
$closed shells
  a      1-4                                    ( 2 )
  t1     1-3                                    ( 2 )
  h      1                                      ( 2 )
$open shells type=1
  a      5                                      ( 1 )
  h      2                                      ( 1 )

Two-electron singlet coupling

The two MOs must have different symmetries (not required for triplet coupling, see example 6.4). We have now two open shells and must specify three sets of (\(a,b\)), i.e. one for each pair of shells, following the keyword $rohf.

Example:

CH\(_2\) in the \(^1\)B\(_2\) state from (3a\(_1\))\(^1\) (1b\(_2\))\(^1\), molecule in (x,z) plane.

$closed shells
  a1      1-2                                    ( 2 )
  b1      1                                      ( 2 )
$open shells type=1
  a1      3                                      ( 1 )
  b2      1                                      ( 1 )
$roothaan         1
$rohf
  3a1-3a1  a = 0      b = 0
  1b2-1b2  a = 0      b = 0
  3a1-1b2  a = 1      b = -2

Two open shells

This becomes tricky in general and we give only the most important case:

shell 1

is a Roothaan case, see 6.4.2

shell 2

is one electron in an a (s) MO \((n_{ir}=1)\)

with parallel spin coupling of shells.

This covers e.g. the \(p^5\,s^1\) \(^3\)P states, or the \(d^4s^1\) \(^6\)D states of atoms. The coupling information is given following the keyword $rohf. The (\(a,b\)) within a shell are taken from above (6.4.2), the cross term (shell 1)–(shell 2) is in this case: \[\begin{align*} a =& 1 \quad \text{always} \\ b =& 2 \quad \text{if} \; n \le n_{ir} \qquad b = \frac{(2n_{ir})}{n} \quad \text{if} \; n > n_{ir} \end{align*}\] where \(n_{ir}\) and \(n\) refer to shell 1.

Example 1:

The \(4d^4\,5s^1\) \(^6\)D state of Nb, in symmetry I

$closed shells
 a       1-4                                    ( 2 )
 t1      1-3                                    ( 2 )
 h       1                                      ( 2 )
$open shells type=1
 a       5                                      ( 1 )
 h       2                                      ( 4/5 )
$roothaan         1
$rohf
5a-5a      a = 0      b = 0
5a-2h      a = 1      b = 2
2h-2h      a = 15/16  b = 15/8

Example 2:

The \(4d^5\,5s^1\) \(^7\)S state of Mo, symmetry \(I\) (see Section 6.4) can also be done as follows.

$roothaan         1
$rohf
5a-5a      a = 0      b = 0
5a-2h      a = 1      b = 2
2h-2h      a = 1      b = 2
$closed shells
 a      1-4                                    ( 2 )
 t1     1-3                                    ( 2 )
 h      1                                      ( 2 )
$open shells type=1
 a      5                                      ( 1 )
 h      2                                      ( 1 )

The shells \(5s\) and \(4d\) have now been made inequivalent. Result is identical to 6.4 which is also more efficient.

Example 3:

The \(4d^9\,5s^1\) \(^3\)D state of Ni, symmetry \(I\)

$closed shells
 a       1-3                                    ( 2 )
 t1      1-2                                    ( 2 )
$open shells type=1
 a       4                                      ( 1 )
 h       1                                      ( 9/5 )
$roothaan         1
$rohf
 4a-4a a = 0     b = 0
 1h-1h a = 80/81 b = 80/81
 4a-1h a =1      b = 10/9

(see basis set catalogue, basis SV.3D requires this input and gives the energy you must get)

6.4.4 Miscellaneous

Valence states

Valence states are defined as the weighted average of all CSFs arising from an electronic configuration (occupation): (MO)\(^n\). This is identical to the average energy of all Slater determinants. \[\begin{equation*} a = b = \frac{2 n_{ir}(n-1)}{(2 n_{ir}-1)n} \end{equation*}\] This covers, e.g. the cases \(n=1\) and \(n=2n_{ir}-1\): \(p^1\), \(p^5\), \(d^1\), \(d^9\), etc, since there is only a single CSF which is identical to the average of configurations.

Totally symmetric singlets for 2 or (2n\(_{ir}\)-2) electrons

\[\begin{alignat*} {2} n &= 2 &\qquad a &= 0 \qquad b = -n_{ir} \\ n &= (2n_{ir}-2) &\qquad a &= \frac{n_{ir}(n_{ir}-2)}{(n_{ir}-1)^2} \\ & & b &= \frac{n_{ir}(n_{ir}-3)}{(n_{ir}-1)^2} \end{alignat*}\] This covers the \(^1\)S states of \(p^2\), \(p^4\), \(d^2\), \(d^8\), etc.

Average of high-spin states: \(n\) electrons in MO with
degenerate \(n_{ir}\).

\[\begin{align*} a &= \frac{n_{ir}\bigl(4k(k+l-1)+l(l-1)\bigr)}{(n_{ir}-1)n^2} \\ b &= \frac{2 n_{ir}\bigl(2k(k+l-1)+l(l-1)\bigr)}{(n_{ir}-1)n^2} \end{align*}\] where: \(k = \max(0,n-n_{ir})\) , \(l = n-2k = 2S \quad\text{(spin)}\)

This covers most of the cases given above. A CSF results only if \(n = \{1, (n_{ir}-1),\) \(n_{ir},\) \((n_{ir}+1),\) \((2n_{ir}-1)\}\) since there is a single high-spin CSF in these cases.

The last equations for \(a\) and \(b\) can be rewritten in many ways, the probably most concise form is

\[\begin{align*} a &= \frac{n(n-2)+2S}{(n-2f)n}\\ b &= \frac{n(n-2)+(2S)^2}{(n-2f)n}\,. \end{align*}\]

This applies to shells with one electron, one hole, the high-spin couplings of half-filled shells and those with one electron more ore less. For \(d^2\), \(d^3\), \(d^7\), and \(d^8\) it represents the (weighted) average of high-spin cases: \(^3\)F + \(^3\)P for \(d^2\),\(d^8\), \(^4\)F + \(^4\)P for \(d^3\), \(d^7\).