XA Report, Introduction "Integral"

Introduction

Calculating the integral

For proving the orthogonality, and therefore the consistency, of data values it is necessary to know the result of the integral of two orbital functions. So both orbital functions are multiplied first, and then the integral is calculated for each part of the sum. The last step can be simplified to the following formula, where

N are the normalization constants for the Clementi orbitals.
c, n and alpha are the data values of the orbitals.

                    n  + n       -r(alpha + alpha )
   oo                1    2              1       2
int    c  c  N  N  r            e                           dr   =
   0    1  2  1  2

                      n  + n
                       1    2                        -1
    =  c  c  N  N  ([r         *  - (alpha  + alpha )            +
        1  2  1  2                        1        2

        - r(alpha  + alpha )
                 1        2   oo       oo                     -1
       e                     ]    - int   (- (alpha  + alpha )   *
                              0        0           1        2

        - r(alpha  + alpha )                 n  + n  - 1
                 1        2                   1    2
       e                       * (n  + n )  r             ) dr)  =
                                   1    2

                                                       -1
    =  c  c  N  N  (  0  +  (n  + n ) (alpha  + alpha )          *
        1  2  1  2            1    2        1        2

                 n  + n  - 1   - r(alpha  + alpha )
          oo      1    2                1        2
       int   (  r             e                       )     dr   =
          0

    =  ...  =

                                 (n  +  n ) !
                                   1     2
    =  c  c  N  N   *  ---------------------------------
        1  2  1  2                         n  + n  + 1
                                            1    2
                         (alpha  +  alpha )
                               1         2

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