Theor Chem Acc (2015) 134:108
1 3
respectively. Now we can substitute Eq. (24) into Eq. ( 21 )
leading to
[In Eq. ( 25B ) the rules for the scalar triple product were
used]. The equations above are valid for any value of p and
q thus our fi nal result is
It can be clearly seen that similar expressions are valid
for the a ni column-elements of the pseudoinverse of the
B -matrix as the b in row-elements of matrix B itself.
3 Conclusion
We prove that the unit vectors associated with the valencetype internal coordinates are not orthogonal generally. It
can be expected that the metrics deduced is suitable for the
acceleration of the obstinate bad convergence of gradient
geometry optimization procedure GDIIS [ 17 ] in case of
very shallow potential surfaces. For instance, such situations are the torsional modes around single bonds (e.g.,
the rotations of methyl groups). The possible gain can be
(24B)
k nq = e q × ρ
0
n ,
(25A)
N
n=1
e p · a ni = 0,
(25B)
N
n=1
e q · ρ
0
n × a ni = 0.
(26A)
N
n=1
a ni = 0,
(26B)
N
n=1
ρ
0
n × a ni = 0 .
signifi cant if using a complete and non-redundant set of
internal coordinates. In our next paper, we will deal with
this project.
Acknowledgments The authors are grateful to Dr. László Párkányi
(Research Centre for Natural Sciences of the Hungarian Academy of
Sciences) and to Dr. Lajos László, Dr. Tibor Pasinszki and János Eőri
(Eötvös Loránd University, Budapest) for the valuable discussions
and help.
References
1. Wilson EB Jr, Decius JC, Cross PC (1955) Molecular vibrations.
McGraw-Hill, New York
2. Boлькeнштeйн MB, Гpибoв ЛA, Eльяшeвич MA, Cтeпaнoв
ςИ (1972) Кoлeбaния мoлeкyл. Hayкa, Mocквa (in Russian)
3. Califano S (1976) Vibrational states. Wiley, London
4. Fogarasi G, Zhou X, Taylor PW, Pulay P (1992) J Am Chem Soc
114:8191
5. Casimir HBG (1931) Dissertation, Leyden
6. Eckart C (1935) Phys Rev 47:552
7. Sayvetz A (1939) J Chem Phys 6:383
8. Eliashevich M (1940) C. R. Acad Sci URSS 28:605
9. Wilson EB Jr (1941) J Chem Phys 9:76
10. Moore EH (1920) Bull Am Math Soc 26:394
11. Penrose R (1955) Proc Cambridge Phil Soc 51:406
12. Pulay P, Fogarasi G, Pang F, Boggs JE (1979) J Am Chem Soc
101:2550
13. Fogarasi G, Pulay P (1985) Ab initio calculation of force fi elds
and vibrational spectra. In: Durig JR (ed) Vibrational spectra and
structure, vol 14. Elsevier, Amsterdam, p 125
14. Crawford B Jr (1952) J Chem Phys 20:977
15. Mirsky L (1990) An introduction to linear algebra. Dover, New
York
16. Pulay P, Fogarasi G, Zhou X, Taylor PW (1990) Vibr Spectrosc
1:159
17. Császár P, Pulay P (1984) J Mol Struct 114:31
18. Malhiot RJ, Ferigle SM (1954) J Chem Phys 22:717
19. Woodward LA (1972) Introduction to the theory of molecular vibrations and vibrational spectroscopy. Oxford University
Press, Oxford
20. Winnewisser BP, Watson JKG (2001) J Mol Spectrosc 205:227
45
Reprinted from the journal
1 3
respectively. Now we can substitute Eq. (24) into Eq. ( 21 )
leading to
[In Eq. ( 25B ) the rules for the scalar triple product were
used]. The equations above are valid for any value of p and
q thus our fi nal result is
It can be clearly seen that similar expressions are valid
for the a ni column-elements of the pseudoinverse of the
B -matrix as the b in row-elements of matrix B itself.
3 Conclusion
We prove that the unit vectors associated with the valencetype internal coordinates are not orthogonal generally. It
can be expected that the metrics deduced is suitable for the
acceleration of the obstinate bad convergence of gradient
geometry optimization procedure GDIIS [ 17 ] in case of
very shallow potential surfaces. For instance, such situations are the torsional modes around single bonds (e.g.,
the rotations of methyl groups). The possible gain can be
(24B)
k nq = e q × ρ
0
n ,
(25A)
N
n=1
e p · a ni = 0,
(25B)
N
n=1
e q · ρ
0
n × a ni = 0.
(26A)
N
n=1
a ni = 0,
(26B)
N
n=1
ρ
0
n × a ni = 0 .
signifi cant if using a complete and non-redundant set of
internal coordinates. In our next paper, we will deal with
this project.
Acknowledgments The authors are grateful to Dr. László Párkányi
(Research Centre for Natural Sciences of the Hungarian Academy of
Sciences) and to Dr. Lajos László, Dr. Tibor Pasinszki and János Eőri
(Eötvös Loránd University, Budapest) for the valuable discussions
and help.
References
1. Wilson EB Jr, Decius JC, Cross PC (1955) Molecular vibrations.
McGraw-Hill, New York
2. Boлькeнштeйн MB, Гpибoв ЛA, Eльяшeвич MA, Cтeпaнoв
ςИ (1972) Кoлeбaния мoлeкyл. Hayкa, Mocквa (in Russian)
3. Califano S (1976) Vibrational states. Wiley, London
4. Fogarasi G, Zhou X, Taylor PW, Pulay P (1992) J Am Chem Soc
114:8191
5. Casimir HBG (1931) Dissertation, Leyden
6. Eckart C (1935) Phys Rev 47:552
7. Sayvetz A (1939) J Chem Phys 6:383
8. Eliashevich M (1940) C. R. Acad Sci URSS 28:605
9. Wilson EB Jr (1941) J Chem Phys 9:76
10. Moore EH (1920) Bull Am Math Soc 26:394
11. Penrose R (1955) Proc Cambridge Phil Soc 51:406
12. Pulay P, Fogarasi G, Pang F, Boggs JE (1979) J Am Chem Soc
101:2550
13. Fogarasi G, Pulay P (1985) Ab initio calculation of force fi elds
and vibrational spectra. In: Durig JR (ed) Vibrational spectra and
structure, vol 14. Elsevier, Amsterdam, p 125
14. Crawford B Jr (1952) J Chem Phys 20:977
15. Mirsky L (1990) An introduction to linear algebra. Dover, New
York
16. Pulay P, Fogarasi G, Zhou X, Taylor PW (1990) Vibr Spectrosc
1:159
17. Császár P, Pulay P (1984) J Mol Struct 114:31
18. Malhiot RJ, Ferigle SM (1954) J Chem Phys 22:717
19. Woodward LA (1972) Introduction to the theory of molecular vibrations and vibrational spectroscopy. Oxford University
Press, Oxford
20. Winnewisser BP, Watson JKG (2001) J Mol Spectrosc 205:227
45
Reprinted from the journal