I don't really understand the "rules" by which the output is generated from the 2 files. Can you explain more about how to generate the output from SI.txt and SI2.txt?

It could be that if you give an example of *.log files, that your desired output could be directly generated from those files without these SI.txt and SI2.txt intermediates? I don't know.

It has been a very,very long time since I used either sed or awk. There is no need for these with Perl. Anyway your grep | awk code is not helpful to me.

Update:
Without really understanding the rules, this code produces your desired output. I guess what I meant above is "tell me what is wrong with this algorithm":

#!/usr/bin/perl
use strict;
use warnings;

## simulate actual files ##

my $SI_txt =<<END;
   Atom      QA       DA(alpha)   DA(beta)    DA(total)
    1      1.0000      2.2238      2.6173      2.3812
    1      1.3294      1.9996      1.9996      1.9996
    2     -0.1098      2.2233      2.2233      2.2233
    3     -0.1098      2.2233      2.2233      2.2233
    4     -0.1098      2.2233      2.2233      2.2233
END

my $SI2_txt =<<END;
Molecule 01-Carbon-Energy.log
Energy: -37.7131454546
Geometry:
Atom  Atomic No.   x                y               z
 1      6       0.000000        0.000000        0.000000
   Atom      QA       DA(alpha)   DA(beta)    DA(total)
Molecule 02-Methanide-Geometry.log
Energy: -39.6946868929
Geometry:
Atom  Atomic No.   x                y               z
 1      6       0.000000        0.000000        0.000000
2       1       0.000000        1.084453        0.000000
3       1       -0.939164       -0.542227       0.000000
4       1       0.939164        -0.542227       0.000000
   Atom      QA       DA(alpha)   DA(beta)    DA(total)
END

###  start of "real code"  ###

open my $SI,  "<", \$SI_txt  or die "unable to open SI.txt";
open my $SI2, "<", \$SI2_txt or die "unable to open SI2.txt";

my @SIarray;
while (<$SI>)
{
   next unless $_ =~ /\d/; #throw away header line or blank
   push @SIarray, $_;
}

while (<$SI2>)
{
    print;
    if (/^\s*Atom\s+QA/) #interleave first line of SI.txt
    {
       print shift @SIarray;
    }
}

print @SIarray;

__END__
Molecule 01-Carbon-Energy.log
Energy: -37.7131454546
Geometry:
Atom  Atomic No.   x                y               z
 1      6       0.000000        0.000000        0.000000
   Atom      QA       DA(alpha)   DA(beta)    DA(total)
    1      1.0000      2.2238      2.6173      2.3812
Molecule 02-Methanide-Geometry.log
Energy: -39.6946868929
Geometry:
Atom  Atomic No.   x                y               z
 1      6       0.000000        0.000000        0.000000
2       1       0.000000        1.084453        0.000000
3       1       -0.939164       -0.542227       0.000000
4       1       0.939164        -0.542227       0.000000
   Atom      QA       DA(alpha)   DA(beta)    DA(total)
    1      1.3294      1.9996      1.9996      1.9996
    2     -0.1098      2.2233      2.2233      2.2233
    3     -0.1098      2.2233      2.2233      2.2233
    4     -0.1098      2.2233      2.2233      2.2233
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