XA Report, Introduction "Creating 1-3"

Introduction

Creating a pseudo orbital

We are now able to create a pseudo orbital. The following example shows the creation process used in Xa:

We assume an atom with two orbitals 1S and 2S. Therefore the next pseudo orbital we are able to create is 3S.

We simplify the orbital function of 3S to:

                     -alpha * r            2     -alpha * r
P    =  c  *  r  *  e            +  c  *  r  *  e            +
 3S      1                           2

               3     -alpha * r
        c  *  r  *  e
         3

By using the precondition of orthogonality:

   oo                       oo                  -alpha * r
int    P   P   dr  =  c  int   P   dr  *  r  * e            +
   0    1S  3S         1    0   1S
                         :...............................:
                                       a
                                        11

                            oo             2    -alpha * r
                      c  int   P   dr  *  r  * e            +
                       2    0   1S
                         :...............................:
                                       a
                                        12

                            oo             3    -alpha * r
                      c  int   P   dr  *  r  * e            =  0
                       3    0   1S
                         :...............................:
                                       x
                                        1

and

   oo                       oo                  -alpha * r
int    P   P   dr  =  c  int   P   dr  *  r  * e            +
   0    2S  3S         1    0   2S
                         :...............................:
                                       a
                                        21

                            oo             2    -alpha * r
                      c  int   P   dr  *  r  * e            +
                       2    0   2S
                         :...............................:
                                       a
                                        22

                            oo             3    -alpha * r
                      c  int   P   dr  *  r  * e            =  0
                       3    0   2S
                         :...............................:
                                       x
                                        2

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