Numerical Hessian for a plannar molecule with n atoms

n atoms:(x1, y1, 0), (x2, y2, 0), (x3, y3, 0), ...

A total of (1 + 9*n^2) energy calculations need to be performed:

1: at equilibrium geometry.
n*1: for 1z-1z 2z-2z,3z-3z,....
[n*(n-1)/2]*2: for 1z-2z, 1z-3z, 1z-4z, 2z-3z, 2z-4z, 3z-4z.
[(2n)^2]*4/2: for the second derivatives with respect to x(s) and y(s)

1 + n + n*(n-1) + 8*n^2 = 1 + 9*n^2

For example, Numerical Hessian for a plannar molecule with 3 atoms

3 atoms:(x1, y1, 0),(x2, y2, 0),(x3, y3, 0)

82 energy calculations need to be performed:

1: at equilibrium geometry.
3: for 1z-1z 2z-2z,3z-3z
3*2: for 1z-2z, 1z-3z, 2z-3z.
2*6^2: for the second derivatives with respect to x(s) and y(s)

1 + 3 + 6 + 72 = 82 (verified by an AM1 method=fullnum Hessian calculation on HOCl)

For a planar molecule such as trans-HNOH with 4 atoms, 1+9*4^2=145 energy calculations need to be carried out to obtain the numerical Hessian.

 $force temp(1) = 1100.00, 1200.00, 1300.00, 1400.00, 1500.00,
                  1600.00, 1700.00, 1800.00, 1900.00, 2000.00
  method=fullnum projct=.t.
 $end
 $contrl maxit=200 mult=2 scftyp=uhf runtyp=hessian coord=unique $end
 $basis gbasis=am1 $end
 $system timlim=1000000 mwords=80 $end
 $data
 HNOH
 Cs

 O           8.0   0.0094634591  -0.0934516366   0.0000000000
 N           7.0   1.2729078083   0.1027207551   0.0000000000
 H           1.0  -0.4112060327   0.7952578868   0.0000000000
 H           1.0   1.7288347653  -0.8045270053   0.0000000000
 $end

For a non-planar  molecule with 4 atoms, see the post on H2NO.

Numeric Hessian for a non-planar H2NO molecule with a Cs Point Group

Both H atoms are bonded to N; the point group is Cs; the symmetry plane contains N-O bond.

$force temp(1) = 10.00, 20.00, 30.00, 40.00, 50.00,
60.00, 70.00, 80.00, 90.00, 100.00
method=fullnum projct=.t.
$end
$contrl maxit=200 mult=2 scftyp=rohf runtyp=hessian coord=unique $end
$contrl ispher=+1 $end
$contrl cctyp=cr-ccl $end
$basis gbasis=cct $end
$system timlim=1000000 mwords=400 memddi=0 $end
$scf soscf=.t. dirscf=.t. $end
$statpt nstep=100 opttol=0.00001 $end
$data
H2NO
Cs

N 7.0 -0.1357158331 0.0890286127 0.0000000000
H 1.0 -0.3822861363 -0.3680309994 0.8687014208
O 8.0 0.0165139267 1.3587083668 0.0000000000
$end

The following 2nd derivatives are evaluated:

Between atoms 1 and 1 9 (2x-x 4x-y 2y-y 1z-z)
Between atoms 1 and 2 (not calculated because 1-2 = 1-3)
Between atoms 1 and 3 (4*3*3=36)
Between atoms 2 and 2 (not calculated because 2-2 = 3-3)
Between atoms 2 and 3 21
Between atoms 3 and 3 18
Between atoms 4 and 1 18
Between atoms 4 and 2 (not calculated because 4-2 = 4-3)
Between atoms 4 and 3 (4*3*3=36)
Between atoms 4 and 4 9

Total = 147 + 1 = 148

Note that due to the Cs symmetry, E(z+dz) = E(z-dz) at z=0


E"(z=0) with respect to z = [E(z+dz) + E(z-dz) - 2E(z)]/dz^2


E"(z=0) = 2[E(z=dz) - E(z=0)]/dz^2


Also, at z=0, E(x+dx,z+dz) = E(x+dx,z-dz);
E(x-dx,z-dz) = E(x-dx,z+dz) due to the Cs symmetry.

Therefore,
E"(z=0) with respect to x and z
= [E(x+dx,z+dz) + E(x-dx,z-dz) - E(x-dx, z+dz) - E(x+dx, z-dz)]/4dxdz = 0

Numerical Hessian

The total number of energy calculations for a N-Cartesian-coordinate system is 2N^2+1.

If N=3n assuming each particle has 3 Cartesian coordinates, then the number of energy calculations = 2*(3n)^2+1 = 18*n^2+1


The single point energy of the geometry is evaluated first.


Then the coordinates are disturbed in the following ways:


For each diagonal element, disturb only 1 coordinate (x) at a time. Evaluate E(x+dx) and E(x-dx). The 2nd derivative can be derived in the following ways:


E'(x+dx/2)=[E(x+dx)-E(x)]/dx


E'(x-dx/2)=[E(x)-E(x-dx)]/dx


E" = [E'(x+dx/2)-E'(x-dx/2)]/dx


E" = [E(x+dx) + E(x-dx) - 2E(x)]/dx^2


For each off-diagonal elements, disturb 2 coordinates (x and y) at the same time. Evaluate E(x-dx,y-dy), E(x-dx,y+dy), E(x+dx,y-dy), and E(x+dx,y+dy).


The 2nd derivative can be derived in the following way:


E'(x+dx,y) with respect to y = [E(x+dx,y+dy)-E(x+dx,E(y-dy))]/2dy


E'(x-dx,y) with respect to y = [E(x-dx,y+dy)-E(x-dx,E(y-dy))]/2dy


E"(x,y) = {[E(x+dx,y+dy)-E(x+dx,E(y-dy))]/2dy - [E(x-dx,y+dy)-E(x-dx,E(y-dy))]/2dy}/2dx


E"(x,y) = {[E(x+dx,y+dy)-E(x+dx,E(y-dy))] - [E(x-dx,y+dy)-E(x-dx,E(y-dy))]}/4dxdy


E" = [E(x+dx,y+dy) + E(x-dx,y-dy) - E(x-dx, y+dy) - E(x+dx, y-dy)]/4dxdy


Note that E(x,y) is not used in the above off-diagonal element calculation.


In summary, 2N+1 energy calculations need to be done to obtain the N diagonal elements of the energy second derivative matrix (Hessian); 4[N(N-1)/2] energy calculations need to be done to obtain the N(N-1)/2 off-diagonal elements of the Hessian matrix.


2N+1+4[N(N-1)/2]=2N^2+1


For example, a 4-particle system has 12 Cartesian coordinates. 2(12^2)+1=289 energy calculations need to be carried out to obtain the Hessian matrix. A 5-particle system has 15 Cartesian coordinates. 2(15^2)+1=451 energy calculations need to be carried out to obtain the Hessian matrix.


The use of the internal coordinates reduces the number of coordinates by 5 or 6. A linear system has 3n-5 internal coordinates while a non-linear system has 3n-6 internal coordinates, where n is the number of particles.


The number of coordinates can be further reduced if the studied system has a symmetry element.

IRC on one of multiple imaginary frequencies

FREQ = The magnitude of the imaginary frequency, given
in cm**-1.
CMODE = An array of the components of the normal mode
whose frequency is imaginary, in Cartesian
coordinates. Be careful with the signs!

You must give FREQ and CMODE if you don't give a $HESS
group, when SADDLE=.TRUE. The option of giving these
two variables instead of a $HESS does not apply to the
GS2 method, which must have a hessian input, even for
restarts.

RK4 may work.

Install Fedora, configure network, and set up GAMESS

Install Fedora-9-x86_64 on Poweredge1950/2950

4xPoweredge 1950: Xeon L5420 16GB memory 450GB HD
1xPoweredge 2950: Xeon L5240 16GB memory 450GB HD + RAID1

Details on Xeon
http://en.wikipedia.org/wiki/Xeon

Download Fedora-9-x86_64 (ISO file for DVD) from anl.gov mirror site (dead) of Fedora
Where to get Fedora 7-10 at 300-350kB/sec
http://mirror.anl.gov/pub/fedora/linux/releases/
I downloaded Fedora 7, 8, 9 at the same time, each at the speed of 300-350kB/s
http://archives.fedoraproject.org/pub/archive/fedora/linux/releases/9/Fedora/x86_64/iso/

Linux 2.6.25 kernel
Restart computer with DVD in the drive
Install or upgrade an existing system: graphics mode is default
"Loading vmlinuz... "
"Anaconda 11.4.0.82 installer"
Media test if the DVD is used the first time
Installation language: English
Keyboard: English
eth0,1,2,3 active on boot; automatically via DHCP (can be revised later)
Time: US Los Angeles CA; uncheck UTC
Enter root password
Remove all linux partitions (default)
Format disk
Select "software development"; uncheck "Office..."; "customize later"(default)
A total of 1110 or 1111 packages will be installed in ~20 minutes
Note the gnome is the default desktop
gcc with gfortran will be installed
Installation done and DVD will be rejected
Reboot
Welcome --> License --> Create user account --> Current time --> Do not send profile
Done with localhost.localdomain

Log in as root (for convenience)

(the following need to be done only 1 time)
(Open firefox --> log in CWU network --> shutdown --> wait 5 minutes --> reboot)

Installation of softwares:
yum install ncompress
yum install csh
yum install tcsh
The above 2 commands install the same tcsh-6.15-4.fc9.x86_64 package

yum provides libstdc++.so.5 (this is for installation of Intel Fortran Compiler Professional)

yum install compat-libstdc++-33
yum install compat-libstdc++-33.i386
Both of the above packages need to be installed for Intel Fortran compiler

Now increase shmmax and shmall
vi /etc/sysctl.conf

# added by Yingbin Ge for 16 GB of SysV memory
kernel.shmmax = 17179869184
kernel.shmall = 4194304

Linux allows this parameter to be set on the fly, by
/sbin/sysctl -w kernel.shmmax=17179869184
/sbin/sysctl -w kernel.shmall=4194304
to avoid a reboot.

Temporarily change selinux=enforcing to selinux=permissive by
vi /etc/sysconfig/selinux
Change selinux=enforcing after the installation of Intel Fortran Compiler (ifort)

reboot; log in as root

Google "Intel Fortran Compiler" and download l_cprof_p_11.0.74.intel64.tgz from Intel website. It includes ifort, debugger and MKL (math kernel lib)
It is free for non-commercial use. Do not close the page with License number.
download l_cprof_p_11.0.74.intel64.tgz to /root/download

gunzip *gz and tar -xvf *tar
cd l_cprof_p_11.0.74.intel64
./install.sh
follow the instructions to install ifort
set path=(. $path /opt/intel/Compiler/11.0/074/bin/intel64)

log out root and log in as regular user
double check network
csh
make a .cshrc file same as /root/.cshrc
source ~/.cshrc

Obtain /u1/guest/gamess.tar.Z and tinker.tar.Z from si.msg
uncompress *.Z
tar xvf gamess.tar
No need to install tinker unless SIMOMM calculations need to be done
cd gamess
cd ddi
vi compddi
make changes
./compddi
mv ddikick.x ..

cd ../tools
cp actvte.code actvte.f
vi actvte.f
:%s/\*UNX/4SPC/g
:wq
ifort actvte.f -o actvte.x
cd ..

vi comp compall lked
set platform = linux-ia64 to use ifort
change paths
compall >& compall.log
lked

vi rungms runall
change paths to /home/yingbin
mkdir ~/scr
su
mkdir /scr
mkdir /scr/yingbin
chown yingbin !$
chgrp yingbin !$
exit

vi /etc/hosts
make changes to combine ip and hostname
127.0.0.1 hostname will do

rungms exam23 failed in the middle due to convergence problem; all others from 1-43 succeeded.

Configure network:
http://www.yolinux.com/TUTORIALS/LinuxTutorialNetworking.html

Log in as root and run .ssh-keygen on each slave node; the produced id_rsa.pub should be appended to the authorized_keys in the .ssh directory of the head node. This is to allow ssh without passwd as a superuser. 
NFS on Fedora 9 (PowerEdge 2950 + 1950)

ON THE HEAD NODE (192.168.2.101 or 192.168.2.102)

1. vi /etc/exports
TYPE the following compute nodes in /etc/exports.

# Allow rw access from compute nodes to server in a synchronized mode.
# Since no_root_squash is put in the parentheses,
# the remote root user will not be treated as a root but as a default nfs user.
/home 192.168.2.111(rw,sync,no_root_squash)
/home 192.168.2.112(rw,sync,no_root_squash)
/home 192.168.2.121(rw,sync,no_root_squash)
/home 192.168.2.122(rw,sync,no_root_squash)
/home 192.168.2.131(rw,sync,no_root_squash)
/home 192.168.2.132(rw,sync,no_root_squash)
/home 192.168.2.141(rw,sync,no_root_squash)
/home 192.168.2.142(rw,sync,no_root_squash)
/home 192.168.2.151(rw,sync,no_root_squash)
/home 192.168.2.152(rw,sync,no_root_squash)

2. /usr/sbin/exportfs -a
This command is to export file system as described in /etc/exports.

3. service nfs restart; service nfslock restart
These commands are to restart nfs and nfslock.

4. chkconfig nfs on; chkconfig nfslock on
These commands are to auto-run nfs and nfslock after reboot.

5. vi /etc/hosts.allow
ALL: .cwu.edu
ALL: 127.0.0.1
This /etc/hosts.allow file is to allow all access from *.cwu.edu
The second line is important to allow loopback to start NFS server.

6. vi /etc/hosts.deny
ALL: ALL
This /etc/hosts.deny file is to deny all access from all IP except from *.cwu.edu

ON EVERY COMPUTE NODE

7. mount -t nfs ge1:/home /home
This command is to mount the ge1:/home on server to a local directory on the compute node.
Then add this following line in /etc/fstab to let the compute node mount master automatically after reboot.
ge1:/home /home nfs hard,intr,rw 0 0

/etc/fstab:
< partition >< mount point >< file system >< mount options >< dump >< fsck order >

/etc/mtab handles the mounted devices and is automatically updated by the mount command. And it looks a bit similar to fstab but not the 100% same ( notice rw and ro for read/write and read only ) And it does only lists the mounted devices !

8. /sbin/chkconfig sendmail off (for security reasons)
/sbin/chkconfig network on (for network on reboot)

9. Set selinux=enforcing on the head node and selinux=disabled on the compute nodes
On the head node: chkconfig iptables on (to turn on the firewall) remember to allow "ssh" using the admin tools on the head node.
On the compute node: chkconfig iptables off

10. to reboot all computers
ssh ge1
su
reboot
ssh ge1
su
ssh na1
reboot
ssh mg1
reboot
ssh al1
reboot
ssh si1
reboot
ssh p1
reboot
 
How to create a user account (yyyy) to run GAMESS

su
useradd yyyy
passwd yyyy

chsh -s /bin/csh yyyy
chown yyyy /home/yyyy
chgrp yingbin /home/yyyy
mkdir /scr/yyyy
chown yyyy /scr/yyyy
chgrp yingbin /scr/yyyy

ssh yyyy@ge1
cp ~yingbin/.cshrc .
vi .cshrc
ssh-keygen
cd .ssh
cp id_rsa.pub authorized_keys
ssh yyyy@ge1
ln -s ~yingbin/gamess ~yyyy/gamess

ssh yyyy@p1
mkdir /scr/yyyy
chown yyyy /scr/yyyy
chgrp yingbin /scr/yyyy