### Generic (non-trigonometric) Rotation of Atomic Orbitals

Normally, a rotation about a diatomic axis does not cause problems when evaluating unitary matrices representing atomic orbitals, but in some instances the phase relationship must be locked.  When this happens, the normal trigonometric analysis cannot be used, and a completely general rotation method must be employed.  This starts with replacing the trigonometric representation of the p-orbitals by a matrix of the type:

 Angular Dependence of the p-orbitals p(x) p(y) p(z) π+ a b c π- d e f σ g h o

where the quantities a-o form a unitary matrix.  Where this matrix comes from depends on the specific system being modeled.  Given that this matrix exists, it can then be used in constructing the rotational properties of the d-orbitals, again in a completely general way.  To help understand the construction of the d-orbital transform, a worked exercise for the first element of the matrix will now be given.  As far as possible, this analysis follows in the same style as the trigonometric analysis.

Using the tools described earlier, let us examine the rotation of d(x2-y2). The rotation components involved are: R(x) = (a)x + (b)y + (c)z,  and R(y) = (d)x + (e)y + (f)z .

The rotation matrix element <d(x2-y2)|R|d(x2-y2)> can be derived using the relationship:
 R|d(x2-y2)> = (a2)x2 + (2a.b)xy + (2a.c)xz + (b2)y2 + (2b.c)yz + (c2)z2 -((d2)x2 + (2d.e)xy + (2d.f)xz +(e2)y2 + (2e.f)yz +(f2)z2) = (a2-d2)x2 + (2(a.b-d.e))xy + (2(a.c-d.f))xz + (b2-e2)y2 + (2(b.c-e.f))yz + (c2-f2)z2

In this expression, all integrals involving odd terms vanish, therefore:

 = + + - - -
The integral <x4> has the same value as <y4>, and <x2y2> has the same value as <x2z2>, etc., therefore this expression can be simplified. Collecting together terms of the type <x4> gives

(a2 - d2 - b2 + e2)

Collecting terms of the type <y2x2> gives

(b2 - e2 + c2 - f2 - a2 + d2 - c2 + f2) = -(a2 - d2 - b2 + e2)

The rotation matrix element is thus:

<d(x2 - y2)|R|d(x2 - y2)> = <x4 -2x2y2 +y4>(a2 - d2 - b2 + e2)

As a result of the fact that the angular components of the atomic orbitals are normalized, i.e., <x4 - 2x2y2 + y4>=1:

<d(x2 - y2)|R|d(x2 - y2)> = (a2 - d2 - b2 + e2)/2

which is the form used in the calculation of the d-rotation matrix. The full set of d-orbital rotation matrix elements is:

 Angular Dependence of the d-orbitals δ+ ≡ d(x2 - y2) π+ ≡ d(xz) σ ≡ d(2z2-x2-y2) π- ≡ d(yz) δ- ≡ d(xy) d(x2-y2) (a2 - d2 - b2 + e2)/2 a.g - b.h (2g2 - d2 - a2 - 2h2 + e2 + b2)/√12 d.g - e.h a.d - b.e d(xz) (a.c - d.f) a.o + c.g (2g.o - d.f - a.c)/√3 d.o + f.g a.f + c.d d(2z2-x2-y2) (2c2 - b2 - a2 - 2f2 + e2 + d2)/√12 (2c.o - b.h - a.g)/√3 (4o2 - 2(h2 + g2 + f2 + c2) + e2 + d2 + b2 + a2)/6 (2o.f - e.h - d.g) /√3 (2c.f - e.b - a.d) /√3 d(yz) (b.c - e.f) b.o + c.h (2h.o - e.f - c.b) /√3 e.o + f.h b.f + c.e d(xy) (a.b - d.e) a.h + b.g (2g.h - e.d - a.b)/√3 d.h + e.g a.e + b.d

The other elements in this matrix can be derived using the following terms:

 R|d(x2-y2)> = (a2)x2 + (2a.b)xy + (2a.c)xz +(b2)y2 + (2b.c)yz + (c2)z2 - (d2)x2 - (2d.e)xy - (2d.f)xz -(e2)y2 - (2e.f)yz -(f2)z2 = (a2-d2)x2 + (2(a.b-d.e))xy + (2(a.c-d.f))xz +(b2-e2)y2 + (2(b.c-e.f))yz + (c2-f2)z2 R|d(xz)> = (a.g)x2 + (a.h + b.g)xy + (a.o + c.g)xz + (b.h)y2 + (b.o + c.h)yz + (c.o)z2 R|d(2z2-x2-y2)> = (2g2)x2 + (4g.h)xy + (4g.o)xz + (2h2)y2 + (4ho)yz + (2o2)z2 - (a2)x2 - (2a.b)xy - (2a.c)xz - (b2)y2 - (2b.c)yz - (c2)z2 - (d2)x2 - (2d.e)xy - (2d.f)xz - (e2)y2 - (2e.f)yz - f2)z2 = (2g2 - a2 - d2)x2 + (4g.h - 2a.b - 2d.e)xy + (4g.o - 2a.c - 2d.f)xz + (2h2 - b2- e2)y2 + (4ho - 2b.c - 2e.f)yz + (2o2 - c2 -f2)z2 R|d(yz)> = (d.g)x2 + (d.h + e.g)xy + (d.o+ f.g)xz + (e.h)y2 + (e.o + f.h)yz + (f.o)z2 R|d(xy)> = (a.d)x2 + (a.e + b.d)xy + (a.f + c.d)xz + (b.e)y2 + (b.f + c.e)yz + (c.f)z2