Anorganische Chemie VI: Theoretical Models and Concepts in Inorganic Chemistry

1
Optimizations

• one of basic tasks in all QC programs
• search for minima, maxima, more general
  stationary points
• characteristics first derivative zero

• classified according to matrix of second
• derivatives (Hessian)

• spectrum of H S(H) all eigenvalues of
• H(x0)

If S(H) lt 0 maximum If S(H) gtlt 0 saddle point
(transition state of order
n) If S(H) gt 0 minimum
2
Optimizations
Optimization Algorithms Non-gradient techniques

• bisection method
• simplex
• genetic algorithms
• etc

• Gradient techniques

Expand function around xo into Taylor series

• Steepest descent

Update parameter vector by a multiple of
the gradient vector
xn1 xn - agn
If a lt 0 walk towards a maximum If a gt 0 walk
towards a minimum

• robust, slow (especially close to minimum)
• avoids saddle points

3
Conjugate Gradients

• Conjugate Gradients (CG)

• Idea restrict search direction to paths that
  are
• orthogonal to all previous searches
• method for solving linear systems of equations
• ? converges to exact solution in no more than N
• steps
• preconditioned CG approximation Ha for H

Ha-1g b
xn1 xn - lnhn

• propagation according to

• directions hn are given by

If n 0
otherwise
where gn g (xn)
and (Fletcher-Reeves variant)
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Conjugate Gradients
or (Polak-Ribière variant)

• step length ln calculated via search along
  direction hn

• Variable Metric Methods
• Newton-Raphson and quasi-Newton

Expand function around stationary point xo into
Taylor series (we make use of the fact that
gradient is zero at x0)

• gradient at x can be expanded too

5
Quasi-Newton
Solving this equation for x0 gives

• for quadratic surfaces and exactly known H,
  this method converges in 1 step
• for non-quadratic surfaces and exactly known H
  (Newton-Raphson) method converges quadratically
• for approximate H this method is called
  quasi-Newton method
• this algorithm converges to nearest stationary
  point (e.g. minimum or saddle point)

Update algorithms for H

• approximations for H can be improved by making
  use of exact first
• derivatives g(xn) at a series of points xn
• several different methods

• Davidon-Fletcher-Powell
• Broyden-Fletcher-Goldfarb-
• Shanno (BFGS)

• Broyden update
• Murtagh-Sargent

6
BFGS

• BFGS updates directly the inverse of H

Convergence Acceleration Techniques
Direct inversion in iterative subspace method
(DIIS)

• P. Pulay, Chem. Phys. Lett. 73, 393 (1980)
• P. Pulay, J. Comput. Chem. 3, 556 (1982)
• P. Csàszàr, J. Molec. Struc. 114, 31 (1984)
• J. Hutter, H.P. Lüthi, M. Parrinello, Comp.
  Mat.. Sci.
• 2, 244 (1993)

7
DIIS

• solves exactly (by direct inversion) an
  optimality condition within the subspace of
  parameter vectors generated by iterations
• lets assume that we have generated a sequence
• of M parameter vectors and
  that we
• are able to guess for each of these m vectors its
  difference em to the stationary point em xm -
  X0
• Ansatz find the best linear combination of
  vectors x

In the ideal case
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DIIS

• this can be fulfilled with

• in real case, we can fulfill the error condition
  only
• approximately, one can write the DIIS equations

Subject to the constraint

• minimization problem can be solved in solved via
• a set of linear equations

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DIIS

• with

• How to approximate the error vectors?

• different possibilities, e.g.

(see last section on quadratic hypersurfaces)
or
or
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(Appendix)
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Numerical Representations (Appendix)

• Decimal system Base10

57368 8 60 300 7000 50000
8 x 100 60 x 101 3 x 102 7 x
103 5 x 104 in general

• Computers Binary system Base2

Just 0 and 1 or on and off 1 bit 1
binary digit, 8bits 1 byte (kilo 103, mega
106, giga 109, tera 1012)

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(57368)10 ( )2?
Example
2 ? 28684, reminder 0 ? c0 0 20 2 ?
14342, reminder 0 ? c1 0 21 2 ? 7171,
reminder 0 ? c2 0 22 2 ? 3585,
reminder 1 ? c3 1 23 2 ? 1792,
reminder 1 ? c4 1 24 2 ? 896,
reminder 0 ? c5 0 25 2 ? 448,
reminder 0 ? c6 0 26 2 ? 224,
reminder 0 ? c7 0 27 2 ? 112,
reminder 0 ? c8 0 28 2 ? 56,
reminder 0 ? c9 0 29 2 ? 28,
reminder 0 ? c10 0 210 2 ? 14,
reminder 0 ? c11 0 211 2 ? 7,
reminder 0 ? c12 0 212 2 ? 3,
reminder 1 ? c13 1 213 2 ? 1,
reminder 1 ? c14 1 214
0, reminder 1 ? c15 1 215
(57368)10 (1110000000011000)2
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Computer Representation
Scientific notation in decimal
57.368 ? 0.57368 x 102 in general x
0.d1d2 x 10n or x r x 10n 1/10 ? r ? 1
sign mantissa exponent
In binary format
x 0.b1b2 x 2n or x r x 2n 1/2 ? r
? 1
? finite number of mantissa digits leads to
round off error
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Subtraction of close numbers
x- sin x (for x close to 0) x
1/15 (radians) x 0.66666 66667 x
10-1 sin x 0.66617 29492 x 10-1
x - sin x 0.00049 37175 x 10-1
0.49371 75000 x 10-4
? avoid subtraction of close numbers (use
e.g. Taylor series)
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Linear Algebraic Equations

• Basic Linear Algebra Subprograms
• (BLAS)

• standard with uniform functionality and
• calling interface
• highly portable across different platforms
• highly optimized

? use BLAS whenever possible!

• BLAS Functions and Subroutines

• vector-scalar operations (BLAS 1)
• matrix-vector operations (BLAS 2)
• matrix-matrix operations (BLAS 3)
• S single precision (32 bit, resp. 64 bit)
• D double precision
• C single precision complex
• Z double precision complex

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Linear Algebraic Equations

• Examples

• DDOT (inner product)
• DAXPY (scale and add)
• DCOPY (copy a vector to another vector)
• DSCAL (scale a vector by a scalar)
• DGEMV (scale a matrix-vector product by a
• scalar and add it to another
  scaled
• vector)
• DGEMM (add a matrix product to another
• matrix)

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Linear Algebraic Equations

• Where can I get BLAS routines?

• Platform specific mathematical libraries
• via NETLIB

• www.netlib.org/ides.html
• anonymous ftp netlib2.cs.utk.edu
• (128.169.92.17)
• email to netlib_at_ornl.org

? source code for all BLAS ? source code for
special functions, spline interpolation and
more ? portable mathematical libraries
LINPACK, EISPACK, LAPACK (for free in
contrast to commercial libraries NAG, ESSL
etc..)
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What are Linear Systems?

• system of m equations with unknowns
• x1..xn
• dependence on x1..xn powers of degree
• ? 1
• summation form

1 ? i ? m

• matrix form

A X B ? X A-1 B
? find inverse of matrix A
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Matrix Inversions

• standard routines in mathematical
• libraries
• Algorithms

• Gauss-Jordan elimination
• Gaussian elimination with
• backsubstitution
• LU Decomposition
• Singular Value Decomposition
• Conjugate Gradients etc..

Gaussian elimination

• generate upper triangular system (forward
  elimination)
• calculate unknowns in reverse order
• (backward substitution)
• pivot row current row being processed
• pivot diagonal element of pivot row
• O(n3) (O(n) for tridiagonal and sparse
• matrices (only few elements ? 0))

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Fast Fourier Transform FFT
Representation of a function in a basis
Basis of harmonic functions
Example Particle in a Box
for 0ltxlta
for xlt0xgta
a
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Fast Fourier Transform FFT
Representation of a function in a basis
Basis of harmonic functions
Example Particle in a Box
for 0ltxlta
for xlt0xgta
a
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Definitions
Fourier Transform (Forward FT)
Inverse FT
? Power Spectrum

• oscillations in time w 2p/T
• oscillations in space k 2p/l
• r,t direct space, time domain
• w reciprocal space or G - space

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Discrete Fourier Transform
Discrete FT

• sampling at discrete time intervals D
• sampling theorem Nyquist Frequency

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Inverse Discrete Fourier Transform

Inverse transform
The formula for the discrete FT can be rewritten
Cooley Tukey
This factorization can be continued until
lengthis 1 ? computational cost reduced from N2
to Nlog(N) Fast Fourier Transform
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3D FFT Applications
3-dimensional FFT
order of operation is interchangeable

• N2N3 1-D FFT of length N1
• N1N3 1-D FFT of length N2
• N1N2 1-D FFT of length N3

Total No. of operations is proportional
to3N2log(N)
Applications of FFT

• signal processing filtering and smoothing
• electronic structure calculations with plane
• wave basis sets
• FT-IR, FT-NMR

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Gaussian Elimination

• Forward Elimination

• generate 0 in columns below diagonal
• process rows downward
• for each row i 1,n-1 // pivot row
• for each row k i 1,n
• multiply pivot row aii aki
• subtract pivot row from row k
• // now column below aii is zero
• // now aij 0

zero pivots! (or values close to zero)
!
? triangular system

• LU Decomposition

• same A many bs (e.g. time-dependent problem)
• want A LU
• define LUx b y Ux
• ? solve Ly b for y
• ? solve Ux y for x

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LU Decomposition

• U is upper triangular (Gaussian elimination)
• L is unit lower triangular (1s on diagonal
• and Gaussian multipliers below)
• A LU
• each new solution only O(n2)

• Singular Value Decomposition

• wi singular values
• U (first n) eigenvalues of AAT
• V eigenvalues of ATA
• ? A-1 V diag(1/wi) UT
• rank number of wi ? 0
• rank n ? A regular otherwise A singular

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Singular Value Decomposition

• for singular matrices cond(A) ?
• ill-conditioned matrices (sensitive to round off
  errors and
• error propagation and magnification)
• 1/cond(A) lt machine precision (typically 10-6
  for single
• precision and 10-12 for double)

• Other means to define numerical
• characteristics Vandermonde Test
• What can be done?

• switch order of equation (move trouble-
• some element off diagonal) pivoting
• guaranteed non-zero pivots
• diagonal dominance

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2. Basic Principles of Quantum
Mechanics
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The Quantum Revolution
Classical Mechanics
Quantum Mechanics
Heisenberg
Planck
Sir Isaac Newton
Erwin Schrödinger Paul A.M. Dirac
(1887 - 1961) (1902 - 1984)
(1642 - 1727)
Nobel Prize in Physics 1933
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Particle Motion
ro,vo ? r(t),v(t)
Position and velocity of a particle can
be calculated exactly at any time t.
Continuous energy
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Waves
Water Waves
Electromagnetic waves
Periodically fluctuating amounts of water
molecules
Sound Waves
Periodical compression of air
2000 Hz
500 Hz
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More Waves
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Wavefunctions
Harmonic oscillation
Hooks Law F - kDq Epo t 1/2
k(Dq)2 w (k/m) 1/2
l wave length (distance from one maximum to
the next) T time period in which the wave
travels 1 l w angular velocity w 2p/T n
frequency (number of periods per second) n
1/T (units 1period/s 1 Hz) c propagation
velocity c l/T ln for electromagnetic
waves c 3x108m/s (velocity of light in
vacuum) for sound waves cs 300 m/s
(velocity of sound in air)
A0
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Wave Propagation
Superposition of Waves
Constructive interference
Destructive interference
Phase shift
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Wave Diffraction
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Summary Particle/Wave Nature
Particles
Waves
e.g. harmonic waves (cos(wt), sin(wt) w2p/T))
long wave length low frequency low energy (Ehn)
l
short wave length high frequency high energy
m,ro,vo,fo ? r(t),v(t) e.g. elastic collisons

• Superimpose (constructive or destructive)
• diffract

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Particle/Wave Duality
Classical Mechanics phenomena can be described
either within a particle or a wave picture
Observation 1 particles can behave like waves!!
P ? (E-V)md-1
Beams of particles can generate
diffraction pattern
Scanning Tunneling Microscope
Particles can tunnel through potential wells
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de Broglie Wavelength
? Particles have also wave character and vice
versa, waves have also particle character!!!
The wave nature is characterized by the de
Broglie wavelength ?
p momentum of the particle h Plancks constant
(h 6.63 x 10-34 Js) m mass of the
particle v velocity of the particle l
wavelength of the particle
Prince Louis de Broglie
(1892-1929) Nobel Prize 1929
(nonrelativistic limit v ltlt c)
Question 1) What is the wavelength of (a) an
electron and (b) a proton which travel at the
same velocity of 1 of the speed of light? 2)
What is your wavelength if you run at a speed of
1m/s?
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Photoelectric Effect
Classical Mechanics The energy spectrum of a
classical object is continuous.
Observation 2 Certain system show a
quantization of the energy!! ? Planck Quantum
Theory
Photoelectric Effect
Classical picture of an atom
Centrifugal Force
Coulomb Force
electron
-
Classically electron could be stabilized at any
distance r with a corresponding velocity v
?
nucleus
? continuos energy spectrum
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Photoeletric Effect
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Spectrum of Atomic Hydrogen
Trifid nebula, constellation Sagittarius, 5200
light years from Earth.The red color arises from
glowing H atoms excited by hot young stars at
the center. The blue glow nearby is light
scattered by dust clouds.
The spectrum of atomic hydrogen. The spectral
lines have been assigned to various groups of
similar wavelength called series the Balmer and
Lyman series are shown here.
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Units atomic units used throughout
Electronic Structure Methods
44
Notation