From m10!frisch@uunet.UU.NET  Thu Nov 26 18:05:05 1992
Message-Id: <9211270419.AA13332@relay1.UU.NET>
Date: Thu, 26 Nov 92 23:05:05 EST
From: m10!frisch@uunet.UU.NET (Michael Frisch)
Subject: Re: GaussianXX orbital orientation
To: chemistry@ccl.net


    
    Does anyone know the orbital orientation that Gaussian90 uses with
    the 5D and 7F orbital options? When using 6D and 10F functions, the
    correct xyz orientations are obvious, but which of xx, yy, zz, 3xx-rr,
    and xx-yy correspond to D0, D+1, D-1, D+2, D-2... ditto for the 
    F orbitals?
    
    This is not as simple a problem to sort out as you may think, and if
    someone knows they could tell me and save me oodles of time...
    
I just replied a day or two ago to a closely realated question sent to me by
someone else, so obviously this is something which should be in the manual.
I'm posting the reply in case there are others with the same question.

The 5 d orbitals are the real combinations of the pure spherical
harmonics (taking the sum and difference of the pure functions
with equal values of |m|).  Specifically, the d functions are:

D0  -- "Z^2" really 3 Z^2 - R^2
D+1 -- XZ
D-1 -- YZ
D+2 -- X^2 - Y^2
D-2 -- XY

The MO coefficients printed are for normalized contracted functions of the
above form.  So 3 of the 5 components map directly to coefficients of
cartesian d functions:

C(X^2) = -1/2 C(D0) + Sqrt(3)/2 C(D+2)
C(Y^2) = -1/2 C(D0) - Sqrt(3)/2 C(D+2)
C(Z^2) = C(D0)
C(XY)  = C(D-2)
C(XZ)  = C(D+1)
C(YZ)  = C(D-1)

where the left-hand sides are the MO coefficients in terms of normalized
cartesian gaussians.

Similarly, for the f functions:

F(0)  = 5 Z^3    - 3 Z R^2
F(+1) = 5 X Z^2  -   X R^2
F(-1) = 5 Y Z^2  -   Y R^2
F(+2) =   Z X^2  -   Z Y^2
F(-2) =   X Y Z
F(+3) =   X^3    - 3 X Y^2
F(-3) = 3 X^2 Y  -     Y^3

or to convert from 7f to 10f:

C(XXX) = -R32Ov2*C(F+1)+R52Ov2*C(F+3)
C(YYY) = -R32Ov2*C(F-1)-R52Ov2*C(F-3)
C(ZZZ) = C(F0)
C(XYY) = -R32Ov2*R5Inv*C(F+1)-TOv2R2*C(F+3)
C(XXY) = -R32Ov2*R5Inv*C(F-1)+TOv2R2*C(F-3)
C(XXZ) = -TOv2R5*C(F0)+R3Ov2*C(F+2)
C(XZZ) = Fact78*C(F+1)
C(YZZ) = Fact78*C(F-1)
C(YYZ) = -TOv2R5*C(F0)-R3Ov2*C(F+2)
C(XYZ) = C(F-2)

Where the constants are
R32Ov2 = Sqrt(3/2) / 2
R52Ov2 = Sqrt(5/2) / 2
Fact78 = 2 Sqrt(3/2) / Sqrt(5)
R3Ov2  = Sqrt(3) / 2
TOv2R2 = 3 / (2 Sqrt(2))
TOv2R5 = 3 / (2 Sqrt(5))


Mike Frisch
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