V - MSpace at the University of Manitoba

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If a complete set (i.e., L = �) were used then every function
24
��
{� i } could be
expressed exactly using this equation. By using a truncated linear combination of
predefined basis functions to express the KS orbitals, however, one reduces the
optimization problem to a much simpler one.
Furthermore, by inserting equation (2–9) into equation (2–8) we obtain an
equation (2–10), very similar to the HF case.
f ˆ
��
KS ��
r 1
L
� ��c�i� �
� �1
��
�r 1��
�i L
��
�c�i��
�r 1�
� �1
(2–10)
Two types of basis functions are used in this research. For g03 and p5 we use
contracted Gaussian–type orbitals (GTO) combined into a contracted Gaussian function
(CGF) of the form:
�
� �
2
GTO a b c ��r
� Nx y z e
(2–11a)
A
�
a
GTO
(2–11b)
CGF � da�� a
Where N is the normalization factor; x, y, and z are the Cartesian coordinates; r is
the radial spherical coordinate; a, b, and c are exponential values that sum to the angular
quantum number l ; α is the orbital exponent which determines compactness (large) or
diffuseness (small) of the function; and dατ are the contraction coefficients.