MODELING MATTER AT
NANOSCALES
3.
3.04.
Empirical classical PES and typical
procedures of optimization
Geometries from energy derivatives
General algorithm: the line
search strategy
An unavoidable pathway for each kind of iterative minimization:
{Rk}k such that k  N E(Z’,Rk+1)  E(Z’,Rk)
is to find optimal step sizes in the way for finding a local or global minimum
of a hypersurface in a given direction.
General algorithm: the line
search strategy
An unavoidable pathway for each kind of iterative minimization:
{Rk}k such that k  N E(Z’,Rk+1)  E(Z’,Rk)
is to find optimal step sizes in the way for finding a local or global minimum
of a hypersurface in a given direction.
It is performed by a search strategy known as line search, meaning:
• Computing E(Z’,Rk) and g(Rk) given Rk at a certain k point of the search.
General algorithm: the line
search strategy
An unavoidable pathway for each kind of iterative minimization:
{Rk}k such that k  N E(Z’,Rk+1)  E(Z’,Rk)
is to find optimal step sizes in the way for finding a local or global minimum
of a hypersurface in a given direction.
It is performed by a search strategy known as line search, meaning:
• Computing E(Z’,Rk) and g(Rk) given Rk at a certain k point of the search.
• Choosing a search direction as F(Rk)
General algorithm: the line
search strategy
An unavoidable pathway for each kind of iterative minimization:
{Rk}k such that k  N E(Z’,Rk+1)  E(Z’,Rk)
is to find optimal step sizes in the way for finding a local or global minimum
of a hypersurface in a given direction.
It is performed by a search strategy known as line search, meaning:
• Computing E(Z’,Rk) and g(Rk) given Rk at a certain k point of the search.
• Choosing a search direction as F(Rk)
• Obtaining step sizes ak > 0 that minimizes E[Z’, Rk + akF(Rk)], i.e. the
hypersurface at a better following location by appropriate mathematical
procedures.
General algorithm: the line
search strategy
An unavoidable pathway for each kind of iterative minimization:
{Rk}k such that k  N E(Z’,Rk+1)  E(Z’,Rk)
is to find optimal step sizes in the way for finding a local or global minimum
of a hypersurface in a given direction.
It is performed by a search strategy known as line search, meaning:
• Computing E(Z’,Rk) and g(Rk) given Rk at a certain k point of the search.
• Choosing a search direction as F(Rk)
• Obtaining step sizes ak > 0 that minimizes E[Z’, Rk + akF(Rk)], i.e. the
hypersurface at a better following location by appropriate mathematical
procedures.
• Selecting the new geometry at the following local step k + 1, i.e. Rk+1 = Rk
+ akF(Rk)
and repeating the cycle from k+1 until achieving a value of a, given a local
convergence threshold that will be the corresponding final optimization step.
First order treatments
Simple gradient methods (first
order): steepest descent
The method of steepest descent only make use of gradient information on
the potential energy surface in order to determine the F direction of search:
F R k    g R k 

Simple gradient methods (first
order): steepest descent
The method of steepest descent only make use of gradient information on
the potential energy surface in order to determine the F direction of search:
F R k    g R k 
Then, each optimization step k + 1 will depend on the previous one following:
R k 1  R k  α k gR k 
Simple gradient methods (first
order): steepest descent
The method of steepest descent only make use of gradient information on
the potential energy surface in order to determine the F direction of search:
F R k    g R k 
Then, each optimization step k + 1 will depend on the previous one following:
R k 1  R k  α k gR k 
The inconvenience is that the gradient at a given point is always
orthogonal to the gradient at the following point in the minimization
 gR k 1   FR k   0
It means a “zig-zag” approach to the minimum that looses much computing
time.
Simple gradient methods (first
order): non – linear conjugate
gradient
The non – linear conjugate gradient method can be considered as a correction
to the method of steepest descent. It remains only making use of gradient
information on the potential energy surface as:
F R 1    g R 1 
Simple gradient methods (first
order): non – linear conjugate
gradient
The non – linear conjugate gradient method can be considered as a correction
to the method of steepest descent. It remains only making use of gradient
information on the potential energy surface as:
F R 1    g R 1 
but subsequent steps behave as:
FR k 1   gR k 1   β k 1FR k 
being bk+1 a matrix of real numbers.
Simple gradient methods (first
order): non – linear conjugate
gradient
The non – linear conjugate gradient method can be considered as a correction
to the method of steepest descent. It remains only making use of gradient
information on the potential energy surface as:
F R 1    g R 1 
but subsequent steps behave as:
FR k 1   gR k 1   β k 1FR k 
being bk+1 a matrix of real numbers.
As well as in the case of steepest descent, each optimization step k + 1 will
depend on the previous one by following:
R k 1  R k  α k gR k 
Simple gradient methods (first
order): non – linear conjugate
gradient
To calculate terms of the bk+1 diagonal matrix in:
FR k 1   gR k 1   β k 1FR k 
several formulas are used depending on gradients themselves:
- as Fletcher - Reeves:
T

g
R k 1 gR k 1 
FR
β k 1  T
g R k gR k 
Simple gradient methods (first
order): non – linear conjugate
gradient
To calculate terms of the bk+1 diagonal matrix in:
FR k 1   gR k 1   β k 1FR k 
several formulas are used depending on gradients themselves:
- as Fletcher - Reeves:
T

g
R k 1 gR k 1 
FR
β k 1  T
g R k gR k 
- or Polak - Ribière:
T

g
R k 1 gR k 1   gR k 
PR
β k 1 
gT R k gR k 
or even other.
Simple gradient methods (first
order): non – linear conjugate
gradient
The big advantage of non – linear gradient conjugate method is that being:
 gR k 1   FR k   0
Simple gradient methods (first
order): non – linear conjugate
gradient
The big advantage of non – linear gradient conjugate method is that being:
 gR k 1   FR k   0
in steepest descent method, here it becomes non - orthogonal:
FR k 1   FR k   0
Simple gradient methods (first
order): non – linear conjugate
gradient
The big advantage of non – linear gradient conjugate method is that being:
 gR k 1   FR k   0
in steepest descent method, here it becomes non - orthogonal:
FR k 1   FR k   0
It means a more straightforward pathway to minima, avoiding zig – zags.
Simple gradient methods (first
order): non – linear conjugate
gradient
It must be observed that non - linear gradient conjugate methods use
the history of gradient development in a certain way. It means that
curvatures (Hessians) are quite accounted.
Advantages are:
• a rather simple formulas for updating the direction vector.
Simple gradient methods (first
order): non – linear conjugate
gradient
It must be observed that non - linear gradient conjugate methods use
the history of gradient development in a certain way. It means that
curvatures (Hessians) are quite accounted.
Advantages are:
• a rather simple formulas for updating the direction vector.
• being slightly more complicated than steepest descent, converges
faster.
Simple gradient methods (first
order): non – linear conjugate
gradient
It must be observed that non - linear gradient conjugate methods use
the history of gradient development in a certain way. It means that
curvatures (Hessians) are quite accounted.
Advantages are:
• a rather simple formulas for updating the direction vector.
• being slightly more complicated than steepest descent, converges
faster.
• more reliable than simpler methods.
Simple gradient methods (first
order): non – linear conjugate
gradient
It must be observed that non - linear gradient conjugate methods use
the history of gradient development in a certain way. It means that
curvatures (Hessians) are quite accounted.
Advantages are:
• a rather simple formulas for updating the direction vector.
• being slightly more complicated than steepest descent, converges
faster.
• more reliable than simpler methods.
It is considered that a good optimization strategy to perform initially a few
steepest descent steps to reach a conformation near to the energy
minimum and then finishing it with conjugate gradient steps until
convergence.
Second order treatments
Gradient and Hessian path to
optimize geometries
The Taylor series can be truncated if Rf is any stationary point of the system or
even being at their nearest surroundings.
Moreover, gradients must collapse at equilibrium or any other stationary
geometry. Then:
1 
E Z ' , R f   E Z ' , R eq   q H R eq q
2
Gradient and Hessian path to
optimize geometries
As the derivative with respect to the position of any a center at geometry Rf is
the gradient of energy at such coordinate, it can also be expanded as a Taylor’s
series:
 E Z ' , R  
g a R f   

ra

R
f
N   2 E Z ' , R  
 E Z ' , R  
 
   
 rbf  rb 
ra

 R b  ra rb  R
Gradient and Hessian path to
optimize geometries
As the derivative with respect to the position of any a center at geometry Rf is
the gradient of energy at such coordinate, it can also be expanded as a Taylor’s
series:
 E Z ' , R  
g a R f   

ra

R
f
N   2 E Z ' , R  
 E Z ' , R  
 
   
 rbf  rb 
ra

 R b  ra rb  R
Therefore, the case of a full gradient matrix at the Rf geometry in matrix
notation is:
g R f   gR   HR q
Gradient and Hessian path to
optimize geometries
When Rf is the equilibrium geometry results that g(Rf) = g(Req) = 0 by
definition, and therefore gradients and curvatures become related by
displacements q as:
gR    HR q
Gradient and Hessian path to
optimize geometries
When Rf is the equilibrium geometry results that g(Rf) = g(Req) = 0 by
definition, and therefore gradients and curvatures become related by
displacements q as:
gR    HR q
Hessian matrix is considered non – singular when HH−1 = H−1H = In,
where In is the identity matrix of n order.
Therefore, the displacement matrix can then be expressed as a product
of the inverse Hessian by the gradient matrix if Hessian is non –
singular:
q  H 1 R gR 
Gradient and Hessian path to
optimize geometries
Therefore, being:
q  H
1
R gR 
whereas q = Req – R, then:
R eq  R  H 1 R gR 
Gradient and Hessian path to
optimize geometries
Therefore, being:
q  H
1
R gR 
whereas q = Req – R, then:
R eq  R  H 1 R gR 
Remembering steps in the line – search procedure:
R k 1  R k  α k gR k 
arises that a good approach for step sizes is the inverse Hessian when it
is non - singular:
H 1 R k   α k
Gradient and Hessian path to
optimize geometries
That means a way to find the equilibrium geometry of any system
following a process where an arbitrary geometry can be progressively
changed downwards in the hypersurface to the point when g(R)
becomes 0 by means of the curvature provided by the Hessian.
Gradient and Hessian path to
optimize geometries
That means a way to find the equilibrium geometry of any system
following a process where an arbitrary geometry can be progressively
changed downwards in the hypersurface to the point when g(R)
becomes 0 by means of the curvature provided by the Hessian.
This is a general principle for finding optimized geometries guided by
the gradient path and the curvature of hypersurfaces at each point.
Gradient and Hessian methods
(second order): Newton Raphson
The Newton – Raphson scheme is based on the principle of using first
derivatives of a function to find extrema and second derivatives to define the
condition of maxima or minima.
Therefore, the general recipe for an iterative term finding the next step is:
g R k 
R k 1  R k 
H R k 
 R k  gR k H 1 R k 
Gradient and Hessian methods
(second order): Newton Raphson
The Newton – Raphson scheme is based on the principle of using first
derivatives of a function to find extrema and second derivatives to define the
condition of maxima or minima.
Therefore, the general recipe for an iterative term finding the next step is:
g R k 
R k 1  R k 
H R k 
 R k  gR k H 1 R k 
Observe that like the steepest descent method, Newton’s searches by the
negative gradient direction.
Gradient and Hessian methods
(second order): Newton Raphson
Newton – Raphson methods provide good output properties and fast
convergence) if started near the equilibrium structure (either local or global).
However, needs modifications if started far away from solution.
Gradient and Hessian methods
(second order): Newton Raphson
Newton – Raphson methods provide good output properties and fast
convergence) if started near the equilibrium structure (either local or global).
However, needs modifications if started far away from solution.
Another inconvenience is that the inverse Hessian is expensive to
calculate.
Gradient and Hessian methods
(second order): Pseudo Newton - Raphson
As not always is possible (as fast as desired) to compute H(Rk) at each
iteration step, there are pseudo – Newton – Raphson methods that use
approximate Hessian matrices:
BR k   H 1 R k 
becoming:
R k 1  R k  BR k gR k 
Being sure that Bk is always definite positive and lim B k  H
k 
1
Gradient and Hessian methods
(second order): Pseudo Newton - Raphson
One case of pseudo – Newton – Raphson procedure is the Broyden – Fletcher
– Goldfarb – Shanno procedure (BFGS), used by default in important program
packages:
BR k q k qTk BR k 
BR k 1   BR k  
qTk BR k q k
T

gR k 1   gR k gR k 1   gR k 

T
gR k 1   gR k  q k
Gradient and Hessian methods
(second order): Pseudo Newton - Raphson
One case of pseudo – Newton – Raphson procedure is the Broyden – Fletcher
– Goldfarb – Shanno procedure (BFGS), used by default in important program
packages:
BR k q k qTk BR k 
BR k 1   BR k  
qTk BR k q k
T

gR k 1   gR k gR k 1   gR k 

T
gR k 1   gR k  q k
The evaluation of B(R1) is performed by particular formulas or by other
method, as those previously described.
Gradient and Hessian methods
(second order): Pseudo Newton - Raphson
Pseudo – Newton – Raphson method use to speed up convergence without
requiring the high computational resources of finding the Hessian matrix at
each optimization step.
Gradient and Hessian methods
(second order): Pseudo Newton - Raphson
Pseudo – Newton – Raphson method use to speed up convergence without
requiring the high computational resources of finding the Hessian matrix at
each optimization step.
In certain cases, as those of large systems with too many degrees of freedom
it could be better to perform steepest descent or conjugate gradient
procedures.
Convergences
There are several convergence criteria which are not
mutually exclusive. If e is a convenient threshold:
→ Mathematical:
gR k   e and all terms in Hk are definite positive
Convergences
There are several convergence criteria which are not
mutually exclusive. If e is a convenient threshold:
→ Mathematical:
gR k   e and all terms in Hk are definite positive
→ Gradient only :
gR k   e although the minimum is not granted
Convergences
There are several convergence criteria which are not
mutually exclusive. If e is a convenient threshold:
→ Mathematical:
gR k   e and all terms in Hk are definite positive
→ Gradient only:
gR k   e although the minimum is not granted
→ Both on gradient and displacements:
gR k   e and R k 1  R k  e
Coordinates
In practice, all matrices are treated in the 3N dimension space of
x, y and z (or x1, x2, x3)on each nuclei or center of reference.
Coordinates
In practice, all matrices are treated in the 3N dimension space of
x, y and z (or x1, x2, x3)on each nuclei or center of reference.
It means that the Taylor series for energy can be expressed in
terms of the X in place of the R matrix:
E Z ' , X f   E Z ' , X  
 x
3N
n 1
f
n
 E Z ' , X  
 xn 
 
 xn
X
2

1 3N 3N f

E Z ' , X  
f



x

x
x

x

 
 n n m m 
2! n1 m1
 xn xm  X
  3 E Z ' , X  
1 3N 3N 3N f
f
f
  ...
   xn  xn xm  xm x p  x p 

3! n1 m1 p 1
 xn xm x p  X
Coordinates
It must be observed that a molecule with 2 atoms have 2N = 6
coordinates, although only one dimension is significant for their
geometry:
x2
x1
x3
x6
x4
x5
→ 3 molecular dimensions (x1, x2, x3) are referred to the
translation of the system with respect to any external point.
Coordinates
It must be observed that a molecule with 2 atoms have 2N = 6
coordinates, although only one dimension is significant for their
geometry:
x2
x1
x3
x6
x4
x5
→ 3 molecular dimensions (x1, x2, x3) are referred to the
translation of the system with respect to any external point.
→ 2 molecular dimensions (x4, x5) are referred to its rotational
motion.
Coordinates
It must be observed that a molecule with 2 atoms have 2N = 6
coordinates, although only one dimension is significant for their
geometry:
x2
x1
x3
x6
x4
x5
→ 3 molecular dimensions (x1, x2, x3) are referred to the
translation of the system with respect to any external point.
→ 2 molecular dimensions (x4, x5) are referred to its rotational
motion.
→ Molecules with more than 2 atoms (non cylindrical symmetry)
require an additional coordinate for dealing with rotations.
Coordinates
To be valid the expression in terms of Cartesian coordinates:
X eq  X  H 1 X gX 
the Hessian must be non – singular, i.e. HH-1 = H-1H = I.
Coordinates
To be valid the expression in terms of Cartesian coordinates:
X eq  X  H 1 X gX 
the Hessian must be non – singular, i.e. HH-1 = H-1H = I.
The problem is that a 3N Hessian matrix in terms of Cartesian
coordinates is singular as well as translation (involving x1, x2, x3)
and rotation (involving x4, x5, x6) of the complete system is a
constant sum of an overall kinetic energy to all terms in Hessian.
It is comprehensive to the whole nanoscopic system.
Coordinates
The system´s geometry deals with the potential energy among
the components that individually regards all other
coordinates.
Coordinates
The system´s geometry deals with the potential energy among
the components that individually regards all other
coordinates.
Therefore, there are no potential energy gradient, nor
Hessian, for the whole system´s translational nor rotational
energy components.
Coordinates
The solution is transforming the full X Cartesian coordinate
matrix in an internal or a reduced Cartesian coordinate matrix
X* with only the geometry significant 3N – 6 terms (or 3N – 5 in
the case of cylindrical systems):
X *eq  X *  H 1 X *gX *
Coordinates
The solution is transforming the full X Cartesian coordinate
matrix in an internal or a reduced Cartesian coordinate matrix
X* with only the geometry significant 3N – 6 terms (or 3N – 5 in
the case of cylindrical systems):
X *eq  X *  H 1 X *gX *
The H(X*) Hessian matrix is non – singular and then all previous
considerations holds for geometry optimizations.
Finding gradients and Hessians
There are two main forms to compute gradients and
Hessians:
– Analytically, meaning the evaluation of formulas of total
energy derivatives from the ones used for the
hypersurface function.
Finding gradients and Hessians
There are two main forms to compute gradients and
Hessians:
– Analytically, meaning the evaluation of formulas of total
energy derivatives from the ones used for the
hypersurface function.
– Numerically, using series to evaluate points and their
corresponding inter- and extrapolations
Finding gradients and Hessians:
numerical approaches
A good algorithm to be used when only energy is known and
analytical gradients can not be obtained is that of Fletcher Powell (FP).
Finding gradients and Hessians:
numerical approaches
A good algorithm to be used when only energy is known and
analytical gradients can not be obtained is that of Fletcher Powell (FP).
It builds up an internal list of gradients by keeping track of the
energy changes from one step to the next. The Fletcher - Powell
algorithm is usually the method of choice when energy
gradients cannot be computed.
Finding gradients and Hessians:
numerical approaches
A good algorithm to be used when only energy is known and
analytical gradients can not be obtained is that of Fletcher Powell (FP).
It builds up an internal list of gradients by keeping track of the
energy changes from one step to the next. The Fletcher - Powell
algorithm is usually the method of choice when energy
gradients cannot be computed.
1. Fletcher, R., A new approach to variable metric algorithms. The Computer Journal
1970, 13 (3), 317-322.
2. Fletcher, R.; Powell, M. J. D., A Rapidly Convergent Descent Method for
Minimization. The Computer Journal 1963, 6 (2), 163-168.
Finding gradients and Hessians:
numerical approaches
Some of the most efficient algorithms are the so – called
quasi-Newton algorithms, which assume a quadratic
potential surface near to minima.
Finding gradients and Hessians:
numerical approaches
Some of the most efficient algorithms are the so – called
quasi-Newton algorithms, which assume a quadratic
potential surface near to minima.
The Berny algorithm internally builds up a second derivative
Hessian matrix.
Finding gradients and Hessians:
numeric and analytic
approaches
Another good algorithm is the geometric direct inversion of
the iterative subspace (GDIIS) algorithm.
Finding gradients and Hessians:
numeric and analytic
approaches
Another good algorithm is the geometric direct inversion of
the iterative subspace (GDIIS) algorithm.
Molecular mechanics programs often use:
Finding gradients and Hessians:
numeric and analytic
approaches
The general DIIS procedure used in GAMESS that includes
both numerical and analytical derivatives is described in the
series:
• Hamilton, T. P.; Pulay, P., Direct inversion in the iterative
subspace (DIIS) optimization of open-shell, excited-state,
and small multiconfiguration SCF wave functions. J.
Chem. Phys. 1986, 84 (10), 5728-5734.
• Pulay, P., Improved SCF convergence acceleration. J.
Comput. Chem. 1982, 3 (4), 556-560.
• Pulay, P., Convergence acceleration of iterative
sequences. The case of SCF iteration. Chem. Phys. Lett.
1980, 73 (2), 393-398.
Finding gradients and Hessians:
refinements near minima and
transition states
The Newton – Raphson approach can be expressed as before
in terms of and internal or reduced Cartesian coordinate
matrix in a stationary point:
q*   H 1 X *gX *
Finding gradients and Hessians:
refinements near minima and
transition states
The Newton – Raphson approach can be expressed as before
in terms of and internal or reduced Cartesian coordinate
matrix in a stationary point:
q*   H 1 X *gX *
For the sake of simplicity it can be written as:
Hq  g  0
Finding gradients and Hessians:
refinements near minima and
transition states
A linear transformation is useful to be performed to treat
variables:
h 0 
 1



H  h  U * HU   0  
  h 

n
Finding gradients and Hessians:
refinements near minima and
transition states
A linear transformation is useful to be performed to treat
variables:
h 0 
 1



H  h  U * HU   0  
  h 

n
Then the corresponding displacements are:
q' U * q
Finding gradients and Hessians:
refinements near minima and
transition states
A linear transformation is useful to be performed to treat
variables:
h 0 
 1



H  h  U * HU   0  
  h 

n
Then the corresponding displacements are:
q' U * q
and gradients:
g ' U * g
Finding gradients and Hessians:
refinements near minima and
transition states
Such hi, q’i and g’i variables are called as the local principal
modes or axes and the Newton – Raphson step along each
principal axis is then:
g 'i
q 'i  
hi
Finding gradients and Hessians:
refinements near minima and
transition states
Such hi, q’i and g’i variables are called as the local principal
modes or axes and the Newton – Raphson step along each
principal axis is then:
g 'i
q 'i  
hi
Then, for stationary points:
g 'i  0
0  h1  h2   hn
0  h1  h2   hn
h1    h  0  h 1   hn
for minima
Finding gradients and Hessians:
refinements near minima and
transition states
Such hi, q’i and g’i variables are called as the local principal
modes or axes and the Newton – Raphson step along each
principal axis is then:
g 'i
q 'i  
hi
Then, for stationary points:
g 'i  0
0  h1  h2   hn
for minima
0  h1  h2   hn
for maxima
h1    h  0  h 1   hn
Finding gradients and Hessians:
refinements near minima and
transition states
Such hi, q’i and g’i variables are called as the local principal
modes or axes and the Newton – Raphson step along each
principal axis is then:
g 'i
q 'i  
hi
Then, for stationary points:
g 'i  0
0  h1  h2   hn
for minima
0  h1  h2   hn
for maxima
h1    h  0  h 1   hn
for a  order saddle point
Finding gradients and Hessians:
refinements near minima and
transition states
It conducts to stepping procedures where:
g 'i
q 'i  
hi  l 
where l is an appropriately chosen shift parameter.
Finding gradients and Hessians:
refinements near minima and
transition states
It conducts to stepping procedures where:
g 'i
q 'i  
hi  l 
where l is an appropriately chosen shift parameter.
Depending upon the value of l, the sign of each (hi - l) will
be positive or negative, and hence the direction of the step
qi will be opposite or toward the direction of the gradient.
Finding gradients and Hessians:
refinements near minima and
transition states
All these expressions facilitate the use of algebra to
determine the best l. These approaches are named as the
eigenvector following procedures.
Finding gradients and Hessians:
refinements near minima and
transition states
All these expressions facilitate the use of algebra to
determine the best l. These approaches are named as the
eigenvector following procedures.
They result excellent to find accurate minima and saddle
points.
Finding gradients and Hessians:
refinements near minima and
transition states
All these expressions facilitate the use of algebra to
determine the best l. These approaches are named as the
eigenvector following procedures.
They result excellent to find accurate minima and saddle
points.
• Banerjee, A.; Adams, N.; Simons, J.; Shepard, R., Search for stationary points on
surfaces. J. Phys. Chem. 1985, 89 (1), 52-57.
• Simons, J.; Joergensen, P.; Taylor, H.; Ozment, J., Walking on potential energy
surfaces. J. Phys. Chem. 1983, 87 (15), 2745-2753.
• Cerjan, C. J.; Miller, W. H., On finding transition states. J. Chem. Phys. 1981, 75 (6),
2800-2806.
Summary
An interesting paper describing how all this matter works in
recent programs is:
Comments
The QM/MM chalenge is also being aforded:
Comments
This is a still active field of research:
References
A comprehensive illustration:
Cramer, C. J., Essentials of Computational Chemistry. Theories
and Models. 2nd. ed.; John Wiley & Sons Ltd: Chichester, 2004; p
596.
An appendix in:
Szabo, A.; Ostlund, N. S., Modern quantum chemistry:
introduction to advanced electronic structure theory. First
edition, revised ed.; McGraw-Hill: New York, 1989; p 466.