Biomedical Engineering Reference
In-Depth Information
Variable
Old X
-DE/DX
Delta X
Delta X
Delta X
New X
(Linear) (Quad)
(Total)
R1
1.90188 -0.00653 -0.00956 0.00000 -0.00956 1.89232
R2
2.73522 -0.00228 -0.00391 0.00000 -0.00391 2.73131
R3
1.90126 -0.00697 -0.01045 0.00000 -0.01045 1.89081
R4
2.05499 -0.00382 -0.00747 0.00000 -0.00747 2.04752
R5
2.89023
0.00948
0.02027 0.00000
0.01759
2.90782
R6
2.90544
0.00539
0.03101 0.00000
0.02505
2.93049
R7
2.29763 -0.05460 -0.03832 0.00000 -0.03832 2.25931
until eventually a stationary point is reached (that is, a point on the molecular potential
energy surface where the gradient is zero). This could be a minimum or a saddle point:
-----------------
! Optimized Parameters !
! (Angstroms and Degrees) !
------------ ------------
! Name Definition Value Derivative Info. !
----------------------------------------
! R1
R(1,2)
1.0012
-DE/DX = 0.0
!
! R2
R(1,3)
1.4507
-DE/DX = 0.0
!
! R3
R(1,4)
1.0003
-DE/DX = 0.0
!
! R4
R(3,5)
1.0824
-DE/DX = 0.0
!
! R5
R(3,6)
1.5232
-DE/DX = 0.0
!
! R6
R(3,7)
1.5386
-DE/DX = 0.0
!
! R7
R(6,8)
1.185
-DE/DX = 0.0
!
...
...
and so on.
17.5 Vibrational Analysis
It is wise to calculate the force constants at this point on the surface, in order to charac-
terize the stationary point. Just to remind you, harmonic force constants correspond to the
eigenvalues of the Hessian, calculated at the stationary point. A minimum on the molecu-
lar potential energy curve should have 3
N
- 6 positive eigenvalues (3
N
5 for a linear
molecule), a transition state of the chemical kind will have just one negative eigenvalue of
the Hessian and so on. In this example there are 23 nuclei and so 69 vibrational coordinates
in total. This gives 63 vibrational coordinates but because I have chosen to do the calcu-
lation in redundant internal coordinates rather than starting from a Z-matrix, there should
be six redundant coordinates corresponding to the three translational coordinates of the
centre of mass and three for the rotations. Each of these should have a force constant of
zero, but there is a question as to how small a number needs to be before it is taken to be
nonzero. This has to do with the optimization cut-off point, the convergence criterion for
the HF-LCAO calculation and so on.
Here is what I found.
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