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6.3 Restricted Open-Shell Hartree–Fock

6.3.1 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 eg2) 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]singlet case. As a guide for expert users, complete ROHF TURBOMOLE input for O2 for various CSFs (configuration state function) is given in Section 21.6. 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):

E = 2∑ khkk + ∑ k,l(2Jkl - Kkl)
+ f[2∑ mhmm + f ∑ m,n(2aJmn - bKmn) + 2∑ k,m(2Jkm - Kkm)]
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.

6.3.2 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 nir. Note that not all cases are Roothaan cases.

All single electron cases are described by:

a = b = 0

Table 6.1: Roothaan-coefficients a and b for cases with degenerate orbitals.
nir=2:  e (div. groups), π, δ (C∞v, D∞h)
n f en πn δn a b
3A 3Σ 3Σ 1 2
2 1∕2 1E* 1Δ 1Γ 1∕2 0
1A 1Σ 1Σ 0 -2
3 3∕4 2E 2Π 2Δ 8∕9 8∕9
1 nir=3:  p (O(3)), t (T, O, I)†
       
n f
pn
 a b
3P
 3∕4 3∕2
2 1∕3
1D**
 9∕20 -3∕10
1S
 0 -3
4S
 1 2
3 1∕2
2D**
 4∕5 4∕5
2P
 2∕3 0
3P
 15∕16 9∕8
4 2∕3
1D**
 69∕80 27∕40
1S
 3∕4 0
5 5∕6
2P
 24∕25 24∕25
only irrep g(I)
(mainly high spin available)
       
n f
gn
 a b
1 1∕8
2G
 0 0
2 1∕4
††
 2∕3 4∕3
1A
 0 -4
3 3∕8
4G
 8∕9 16∕9
4 1∕2
5A
 1 2
5 5∕8
4G
 24∕25 32∕25
6 3∕4
††
 26∕27 28∕27
1A
 8∕9 4∕9
7 7∕8
2G
 48∕49 48∕49
d(O3), h(I)
(mainly high-spin cases work)
       
n f
dn
 a b
1 1∕10
2D
 0 0
2 1∕5
3F+3P††
 5∕8 5∕4
1S
 0 -5
3 3∕10
4F+4P††
 5∕6 5∕3
4 2∕5
5D, 5H
 15∕16 15∕8
5 1∕2
6S, 6A
 1 2
6 3∕5
5D, 5H
 35∕36 25∕18
7 7∕10
4F+4P††
 95∕98 55∕49
8 4∕5
3F+3P††
 125∕128 65∕64
1S
 15∕16 5∕8
9 9∕10
2D, 2H
 80∕81 80∕81
* except cases (e.g. D2d or D4h) where e2 gives only one-dimensional irreps, which are not Roothaan cases.
† only pn given, the state for groups Td etc. follows from S → A (T,O,I) P → T (T,O,I) D → H (I), E+T (T,O)
** This is not a CSF in T or O, (a,b) describes average of states resulting from E+T
†† (a,b) describes weighted average of high spin states, not a CSF.
Example

The 4d95s2 2D 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

6.3.3 More Than One Open Shell

A Half-filled shell and all spins parallel

All open shells are collected in a single open shell and

a = 1 b = 2

Example: The 4d55s1 7S 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 6.3.3). 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: CH2 in the 1B2 state from (3a1)1 (1b2)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 6.3.2
shell 2
is one electron in an a (s) MO (nir = 1)

with parallel spin coupling of shells.

This covers e.g. the p5s1 3P states, or the d4s1 6D states of atoms. The coupling information is given following the keyword $rohf. The (a,b) within a shell are taken from above (6.3.2), the cross term (shell 1)–(shell 2) is in this case:

a = 1 always
b = 2 ifn ≤ nir b = (2nir)- n ifn > nir
where nir and n refer to shell 1.

Example 1: The 4d45s1 6D 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 4d55s1 7S state of Mo, symmetry I (see Section 6.3.3) 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 6.3.3 which is also more efficient.

Example 3: The 4d95s1 3D 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)

6.3.4 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.

a = b = 2nir(n---1) (2nir - 1)n

This covers, e.g. the cases n = 1 and n = 2nir - 1: p1, p5, d1, d9, etc, since there is only a single CSF which is identical to the average of configurations.

Totally symmetric singlets for 2 or (2nir-2) electrons

n = 2 a = 0 b = -nir
n = (2nir - 2) a = nir(nir---2) (nir - 1)2
b = nir(nir---3) (nir - 1)2
This covers the 1S states of p2, p4, d2, d8, etc.

Average of high-spin states: n electrons in MO with
degenerate nir.

a = nir(4k(k-+-l --1)+-l(l --1)) (nir - 1)n2
b = 2nir(2k(k + l - 1)+ l(l - 1)) -----------------2--------- (nir - 1)n
where: k = max(0,n - nir) , l = n - 2k = 2S (spin)

This covers most of the cases given above. A CSF results only if n = {1,(nir - 1), nir, (nir + 1), (2nir - 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

a = n(n---2)+-2S- (n- 2f)n
b = n(n - 2)+ (2S )2 ---(n---2f)n----.
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 d2, d3, d7, and d8 it represents the (weighted) average of high-spin cases: 3F + 3P for d2,d8, 4F + 4P for d3, d7.

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