Computational Chemistry

2. Calculate the integrals:Trs,Vrsfor each nucleus, and the two-electron integrals (ru|
   ts) etc. needed forGrs, as well as the overlap integralsSrsfor the orthogonalizing
   matrix derived fromS(see step 3). Note: in thedirect SCFmethod (Section 5.3)
   the two-electron integrals are calculated as needed, rather than all at once.


3. Calculate the orthogonalizing matrixS#1/2
   (a) DiagonalizeS:S¼PDP#^1
   (b) CalculateD#1/2(take the#1/2 power of the elements ofD)
   (c) CalculateS#1/2¼PD#1/2P#^1


4. Calculate the Fock matrixF
   (a) Calculate the one-electron matrixHcore¼TþV 1 þV 2 þ(((using the
   TandVintegrals from step 2
   (b) The two-electron matrix (the electron repulsion matrix)G:
   Use an initial guess of the coefficients of the occupied MO’s to calculate initial-
   guess density matrix elements:

Ptu¼ 2

Xn

j¼ 1

c$tjcuj t¼ 1 ; 2 ;...;mandu¼ 1 ; 2 ;...;m

Use the density matrix elements and the two-electron integrals to calculateG:

Grs¼

Xm

t¼ 1

Xm

u¼ 1

Ptu ðrsjtuÞ#

1

2

ðrujtsÞ



The Fock matrix isF¼HcoreþG

5. TransformFtoF^0 , the Fock matrix that satisfiesF^0 ¼C^0 eC^0 #^1

F^0 ¼S#^1 =^2 FS#^1 =^2

6. DiagonalizeF^0 to get energy levels and aC^0 matrix

F^0 ¼C^0 eC^0 #^1

7. TransformC^0 toC, the coefficient matrix of the original basis functions

C¼S#^1 =^2 C^0

8. Compare the density matrix elements calculated from theCof the previous step
   with those of the step before that one (and/or use other criteria, e.g. the molecu-
   lar energy); if convergence has not been achieved go back to step 4 and calculate
   a new Fock matrix using theP’s from the latestc’s. If convergence has been
   achieved, stop
   It should be realized modern ab initio programs do not rigidly follow the basic
   SCF procedure described in this section. To speed up calculation they employ a

5.2 The Basic Principles of the ab initio Method 231