OPEN(n₁,n₂)

RHF description: The M.O. occupancy during the SCF calculation can be defined in terms of doubly occupied, empty, and fractionally occupied M.O.s. By default, RHF SCF calculations are run using doubly occupied M.O.s, with, at most, one singly occupied M.O.  For some systems, the symmetry of the M.O.s can be preserved only if two or more M.O.s have fractional occupancies. Such fractional occupancies can be defined using OPEN(n₁,n₂), where n₁ = number of electrons in the open-shell manifold, and n₂ = number of open-shell M.O.s;  The ratio of n₁ to n₂ is restricted to  2 >  n₁/n₂  > 0.  Example: if OPEN(3,4)were used then the occupancy near the HOMO-LUMO gap would be ...2, 2, 2, 0.75, 0.75, 0.75, 0.75, 0, 0, 0, ...

The two π M.O.s in  molecular oxygen would not be degenerate if a closed-shell SCF were run.  In order to maintain the degeneracy, OPEN(2,2) would need to be used.

Do not use OPEN(n₁,n₂) for ground-state systems except for high symmetry systems with open shells, such as twisted (D_2d) ethylene or molecular oxygen O₂, or if there is a very small band-gap (such as in metal clusters).

Examples of OPEN(n₁,n₂) are given in the Table. OPEN(1,1) will be assumed for odd-electron systems unless an OPEN keyword is used. Errors introduced by use of fractional occupancy are automatically corrected [136] in a MECI calculation when OPEN(n₁,n₂) is used. See also C.I.=n.

Notes:

• OPEN(n₁,n₂)
cannot be used instead of C.I.=n₂. OPEN(n₁,n₂) modifies the SCF calculation, but C.I.=n₂ does not modify the SCF calculation. Both keywords cause a C.I. calculation to be done.
• Be careful to ensure that n₁ is odd if the system has an odd number of electrons, and even if the system has an even number of electrons.

UHF description: The M.O. occupancy during the SCF calculation can be defined in terms of singly occupied, empty, and fractionally occupied M.O.s. By default, UHF SCF calculations are run using singly occupied M.O.s. When OPEN(n₁,n₂) is present, the highest n₂ α molecular orbitals are each given a population of n₁/n₂ electrons.  To use OPEN(n₁,n₂) in a UHF calculation, UHF must also be present.  Because fractional electrons are used, small errors in energy, on the order of 1-2 kcal/mol, are introduced when OPEN(n₁,n₂) is used.  These errors are not corrected by C.I., however they are small and should not affect the results significantly.

Because OPEN only applies to the α electrons, care must be exercised in defining the number of α and β electrons.  For example, if the cation of tetrahedral methane were to be calculated, the keywords needed would be "OPEN(2,3)MS=-0.5 UHF CHARGE=1" If MS=-0.5 was not present, then there would be four α and three β electrons, instead of the required three α and four β electrons.

The main use of OPEN(n₁,n₂) in UHF calculations is to mimic the dynamic Jahn-Teller effect in transition metal complexes.  Without OPEN(n₁,n₂)an octahedral complex  system that could undergo Jahn-Teller distortion would be distorted.  However, in many X-ray structures, systems that should Jahn-Teller distort do not in fact distort and instead retain high symmetry, typically O_h.  This high symmetry can be regarded as a dynamic J-T effect. Transition metal complexes are, as their name suggests, complicated, and correctly defining the various J-T states is also complicated.  The following set of examples is provided to illustrate how to define keywords for various octahedral systems.  These systems are assumed to have a three-fold degenerate set of "d" M.O.s, with dominant contributions from the d-orbitals d_xy, d_yz, d_xz, that can be related to the t_2g set of point group O_h, and a two-fold degenerate set, composed mainly of the other two d-orbitals; these are equivalent to the  e_g set of point group O_h.

d⁰: No keyword needed - closed shell, i.e., ¹A_1g.
d¹: keywords: "UHF OPEN(1,3)"  One α d electron in a  t_2g set of M.O.s.  The resulting state is ²T_2g.
d²: keywords: "UHF OPEN(2,3) MS=1"  Two α d electrons in a  t_2g set of M.O.s.  The resulting state is ³T_1g.
d³: keywords: "UHF MS=1.5"  The state is ⁴A_2g.
d⁴: keywords for low spin: "UHF OPEN(1,3)MS=-1"  Because OPEN(n₁,n₂), applies only to α electrons, and because there are four electrons in a  t_2g set of M.O.s, the only way to have a fractional population of the α set of M.O.s is to have the β set filled, i.e., α¹β³ . By using "MS=-1" the number of  β electrons is defined as being two more than the number of α electrons  The resulting state is ³T_1g.
d⁴: keywords for high spin: "UHF OPEN(1,2) MS=2"  In high-spin d⁴ complexes, there are three α electrons in the t_2g  set of M.O.s, and a single α electron in the two-fold degenerate e_g M.O.  "MS=2" results in there being four more α than β electrons.  The resulting state is ⁵E_g.
d⁵: keywords for low spin: "UHF OPEN(2,3) MS=-0.5"  See d⁴ low spin. This gives rise to a  ²T_2g state.
d⁵: keywords for high spin: "UHF MS=2.5"  The state is ⁶A_1g.
d⁶: No keyword needed - closed shell, i.e., ¹A_1g.
d⁶: keywords for high spin: "UHF OPEN(1,3) MS=-2"  The resulting state is ⁵T_2g = A_2g (from the β² of e_g symmetry)*A_2g (from the β³ of t_2g symmetry)*T_2g (from the α¹ of t_2g symmetry).
d⁷: keywords for low spin: "UHF OPEN(1,2)"  As with d¹, the extra electron has, by default, α spin, and so the keyword "MS=0.5" is unnecessary.  The resulting state is ²E_g.
d⁷: keywords for high spin: "UHF OPEN(1,2) MS=-1.5"  With seven d-electrons and  MS=-1.5 there are three more α than β electrons, i.e., α²β⁵.  The resulting state is  ⁴T_1g.
d⁸: keywords: "UHF MS=1"   The resulting state is  ³A_2g.
d⁹: keywords: "UHF OPEN(1,2)"   The resulting state is  ²E_g.
d¹⁰: No keyword needed - closed shell, i.e., ¹A_1g.

A similar exercise could be done with the T_d complexes. As this is simple, albeit tedious, it is left as an exercise.