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# Møller-Plesset calculations for open-shell systems

There are several variants of open-shell Møller-Plesset calculations depending upon whether the preceding SCF calculation is UHF or ROHF (high-spin) in type. We shall consider UHF based methods first.

## UHF methods

The following dataset provides an example.
```TITLE
SYMMETRY
CS
END
BASIS 631G*

ATOMS
C1          6.0    0.00000000     0.00000000     0.00000000
C2          6.0    0.00000000     0.00000000     2.83099850
H1          1.0    1.91452506     0.00000000    -0.76343074
H2          1.0   -0.97037160     1.64604991    -0.74613695
H3          1.0    0.24692716    -1.73500069     3.85917480
END
UHF
MP4
MULTIPLICITY 2
ENERGY
START
FINISH```
This sets up the geometry, in Cartesian coordinates. The SCF type is set to UHF and the multiplicity must also be given. The directives MP4 and ENERGY then specify the level of correlation and the fact that only the energy is required.

Other possible directives are MP2, MP3 and MP4SDQ which give lower levels of correlation. The final output from the dataset given above looks like this :

```    MP2(S+D)=     -0.24727265
MP3(S+D)=     -0.02615266
ESCF    =    -78.59709385
EMP2    =    -78.84436649
EMP3    =    -78.87051915
MP4(S)  =     -0.00172931
MP4(D)  =     -0.00679836
MP4(Q)  =      0.00346152
MP4(T)  =     -0.00541343
EMP4    =    -78.88099872
S-SQUARED =    0.76229266```
The MP2, MP3 and MP4 correlation energies are given separately, together with the appropriate total energies, defined as the sum of all terms up to a given order and including the SCF energy. The spin expectation value (S2) is also given - note that in UHF one does not have a pure spin state. If the values of S2 deviates significantly (more than 10%) from its correct value then UHF based methods are likely to be unreliable.

## ROHF methods

It is also possible to calculate energies based on ROHF wavefunctions. In general an ROHF formalism is to be preferred to UHF as there is no spin-contamination and this avoids certain pathological cases for which UHF gives very bad answers. For a discussion and detailed results see Molec. Phys. 79, 777 (1993).

Consider the following dataset. This is also C2H5, as with the previous example, except that the geometry is specified in internal cooordinates (this is simply to provide an extra example - it is not an essential part of the calculation). The other difference is that the high-spin open-shell SCF program is used for the SCF part of the calculation.

```TITLE
SYMMETRY
CS
END
VARIABLES
RCC  1.4981 A
RCH1 1.0907 A
RCH2 1.0855 A
RCH3 1.0752 A
CCH3 120.40 D
CCH1 111.74 D
CCH2 111.33 D
TOR2 120.52 D
TOR3 81.9 D
END
BASIS 631G*
ATOMS
C1  6     0.0     0.0    0.0
C2  6     0.0     0.0    RCC
X -1 1 0 0
H1  1  PTC C1 C2 X RCH1 CCH1
H2  1  TCT C1 C2 H1 RCH2 CCH2 TOR2
H3  1  TCT C2 C1 H1 RCH3 CCH3 TOR3
END
ROHF
MULTIPLICITY 2

MP2
ENERGY
START
FINISH```
It is also possible to calculate MP3 and MP4 correlation energies based on ROHF (=OSCF) SCF calculations simply by changing MP2 in the example to MP3 or MP4 respectively. Note that the only difference in the dataset as compared to the UHF cases is the keyword which specifies the SCF type.

The output from an MP4 calculation using the dataset given above looks like this :

``` ROHF-MP (RMP) Calculation (See Chem Phys Lett vol 186 page 130)
MP2(S+D)=     -0.25169806
MP3(S+D)=     -0.02633835
ESCF    =    -78.59269803
EMP2    =    -78.84439608
EMP3    =    -78.87073443
MP4(S)  =     -0.00179698
MP4(D)  =     -0.00623789
MP4(Q)  =      0.00353021
MP4(T)  =     -0.00602285
EMP4    =    -78.88126195
S-SQUARED =    0.75000000```
The various terms in the output have the same meanings as in the UHF case. Note that although the SCF energy with ROHF is slightly higher than in the UHF calculation given above, the MP4 energy is slightly lower - this is typical of ROHF-MP calculations. Note also that the value of S2 is exact.

Møller-Plesset calculations based on ROHF orbitals have a degree of arbitrariness in that the results depend on how the SCF orbitals are canonicalised. The results obtained from the previous dataset are based on the method described in Chem. Phys. Lett. 186, 130 (1991). This is probably the most widely used ROHF-MP2 approach. There is an alternative approach described in Chem. Phys. Lett. 185, 256 (1991). To obtain Møller-Plesset energies corresponding to this scheme one uses in place of the keywords MP2, MP3 and MP4 the keywords ROMP2 , ROMP3 or ROMP4 respectively. The results of the two approaches are very similar. For example in the case of the C2H5 structure given above the ROMP4 results are :

``` ROMP Calculation (See Chem Phys Lett vol 185 page 256)
MP2(S+D)=     -0.25169306
MP3(S+D)=     -0.02624420
ESCF    =    -78.59269803
EMP2    =    -78.84439109
EMP3    =    -78.87063529
MP4(S)  =     -0.00183780
MP4(D)  =     -0.00628749
MP4(Q)  =      0.00352380
MP4(T)  =     -0.00603788
EMP4    =    -78.88127466
S-SQUARED =    0.75000000```
Note that the outputs from the two schemes have virtually identical formats, and they are distinguished by having the appropriate reference included as part of the printout, and by the designation ROHF-MP or ROMP.

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