Download Document

Basic Quantum Chemistry: how to
represent molecular electronic states
Jimena D. Gorfinkiel
Department of Physics and Astronomy
University College London
Summary
• I will describe the basic ideas and procedures behind the
determination of wavefunctions and eigenvalues describing the
ground and excited electronic states of polyatomic molecules.
• The talk is not a detailed theoretical analysis nor an exhaustive
listing of methods (of which there are many)
• Ab initio means from first principles. It does not mean exact.
• Semi-empirical methods also available
• Variational methods
• Perturbative methods also available
The Variational Principle
E0 (exact) < E0 (approx)
Variational principle establishes that the expectation value of the
Hamiltonian provides an upper bound to the exact energy  the
lower the energy, the better it is!
It also follows that increasing the number of elements in a basis
will improve (or at least not worsen) the result
• Also applies to excited states
• NOT all electronic structure methods are variational (e.g. MP2,
MP3 etc)
How to describe a molecule
ei
riA
ri defines position of electron i
riB
ri
A
R
si is the spin coordinate of electron i
x indicates centre of mass of system
B
Separating total translation of molecule:
1 2 N 1 2 N Z A N Z B N N 1 Z AZ B
H 
R   r  

 
i 1 2
i 1 riA
i 1 riB
i 1 j  i rij
2
R
i
Hel
Bohr-Oppenheimer Approximation
• Decoupling of electronic and nuclear motion
• Electrons are much lighter hence adapt ‘instantaneously’ to
movement of nuclei (cows and flies)
• More formally, gradients of  with respect to R are neglected
 (r i , R)   (r i ; R) ( R)
R becomes a parameter:  is calculated for a specific set of R values
( can then be calculated using electronic energies as potentials)
We need to solve:
H
el
1 2 N ZA N ZB N N 1
   r  


i 1 2
i 1 riA
i 1 riB
i 1 j  i rij
N
i
How to describe electronic states
H
el
1 2 N ZA N ZB N N 1
   r  


i 1 2
i 1 riA
i 1 riB
i 1 j  i rij
N
i
• Can’t be solved exactly (except for H2+)
• Must use approximate methods
• Implementations make use of molecular symmetry to simplify
numerical calculations
We could write the multielectronic wavefunction as a product of 1particle functions: MOLECULAR METHOD
 = molecular orbital = f(x,y,z)
Spin-orbital =  x spin function = f(x,y,z,s,sz)
 indicates spin +1/2 (or up)
 indicates spin -1/2 (or down)
To obtain a multielelectronic wavefuntion we multiply MOs:
 (r 1 , r 2 , r N )  1 (r1 ) 1 2  2  N  N
2
2
(the example is a closed-shell ground state configuration)
║ ║ are Slater determinants and indicate that the product is
antisymmetric with respect to particle exchange
Symmetry
Molecules belong to a specific point
group. The wavefunctions (total, orbitals,
etc.) will be symmetric or antisymmetric
with respect to applying certain
symmetry elements,
Making use of this symmetry properties
greatly simplifies computational side.
‘Names’ of irreducible representations
are used to label the states (e.g., A’, u,
B3g)
Electronic wavefunctions should be eigenfunctions of the spin operator.
States (and configurations) can then be labelled as singlet, doublet, triplet, etc
(but not always!).
Hartree-Fock approximation
 (r i ; R)  1 (r1 ) 1 2  2  N  N
2
2
We look for those orbitals i
that minimise E
Basic idea is that the effect of the N-1 electrons on Nth electron can
be approximated as an averaged field
Hartree-Fock equation:
F i   i i
N
F  h   (2 J j  K j )
j 1
Jj and Kj are the coulomb and exchange operators and they depend on
all the other orbitals.
*Normally used for ground states
How do we solve the equation if the operators themselves depend on
the orbitals we are trying to obtain?
HF Self Consistent Field method (SCF):
Iterative procedure with initial set of trial orbitals. Equations
are solved until energy obtained in 2 successive iterations is
identical, within some specified tolerance limit.
• Restricted HF: spin-orbitals have same spatial part for spin up and
spin down
• Unrestricted HF: spin-orbitals can have different spatial part for
spin up and spin down. Used for open-shell systems. Problems with
spin contamination.
• Restricted open-shell HF: closed-shell electrons occupy obitals
with same spatial function. Eigenfunctions of spin operator but E is
raised.
Basis sets
• These are the analytical functions in which the 1-particle orbitals
are expanded.
• Normally single-centre and centred on the nuclei (although can be
centred somewhere else)
A variety of functions are used: STOs and GTOs are the most
common but also B-splines, etc. particularly in non-standard
calculations
Slater Type Orbitals (STOs):
f (r )r
•
•
•
•
n 1  r
e
Υ lm ( ,  )
Solutions to the H-atom problem
Correct cusp at the nucleus
Correct exponential long-range behaviour
Integrals must be evaluated numerically, gives
approximately 8 figure accuracy.
general programs only for diatomic (linear) molecules
• Basis sets not widely used / available
Gaussian Type Orbitals (GTOs):
f ( r )e
 r 2
Υ lm ( ,  )
• Finite at the nucleus: no cusp
• long-range decay too fast
• Integrals evaluated analytically (12+ figures)
many, many general programs available
• Systematic series of GTOs available
• Libraries of basis sets available on the web
• give a poor representation of high n Rydberg states
How to chose GTOs?
• How many l,m?
Dictated by number of electrons, polarization, size of
calculation
• What are the right exponents?
Lots of literature available. Sets of exponents for each atom
(sometimes needs adaptation)
L
Contracted GTO:

 Ci ( i )
i 1
Ci are optimized in different ways (and tabulated); using them
makes optimization easier.
(bare in mind: we are not trying to do ‘standard’ quantum chemistry!)
Some examples
 STO-3G: minimal basis set
•1 function per occupied orbital (5 Li to Ne, 9 Na to Ar, etc.)
• 3 GTOs contracted by least square fit to STOs
•  does NOT depend on l (so 2s and 2p have same ’s)
 4-31G: double-zeta (sort of)
• DZ: 2 functions for each of the minimal basis
• valence functions doubled, but single for each inner shell orbital
(2 H and He, 9 Li to Ne, 13 Na to Ar)
• Contractions: 4 GTOs for inner shell, 3 and 1 GTOs for valence
• contraction coefficients and  obtained by minimizing E
 6-31G*, 6-31G** : polarized basis set
• Triple Z not well balanced: better to add l +1 functions (p to H
and d to Li-F)
• *: d to heavy atoms; **: d to heavy atoms and p to H
(uncontracted)
• Contractions: 6 GTOs for inner shell, 3 and 1 GTOs for valence
• contraction coefficients and  obtained by minimizing E;
valence similar to 4-31G but not identical
Diffuse functions: those with small . Important for excited states,
anions, etc..
Molecular Orbitals
• Molecular orbitals are built as linear combination of basis functions
• They are multicentric
• They describe 1-particle
M
 i (r )   aij  (r )
j
aij can be obtained via HF-SCF or by other means (Natural
Orbitals, Improved Virtual Orbitals, etc.)
SCF Orbitals
• Solutions of the Hartree-Fock equations
(usually obtained iteratively using basis sets)
• Problems with dissociation e.g. H2  50% (H + H) + 50% (H+ + H)
• Only optimised for single configuration
(usually the ground state), poor representation of other states
Natural Orbitals
• They give the most rapidly convergent CI expansion (see later)
• Obtained diagonalizing the one-electron reduced density matrix
• Associated eigenvalue is not an energy but an occupation
number
Configurations
To obtain a multielelectronic wavefuntion we multiply MOs:
 (r 1 , r 2 , r N )  1 (r1 ) 1 2  2  N  N
2
2
The product will have a defined space-spin symmetry
Which orbitals we multiple and how many configurations
we build will be discussed in the next talk.
Configuration Interaction
A single-configuration representation is not good enough in most
cases because:
• Orbitals generated with a HF-SCF method are best to represent
ground state
• Even in this case, a single configuration cannot represent
correlation
Correlation: ‘electrons move in such a way that they keep more
apart from each other than close’
Ecorr=Eexact- EHF-limit
increasing number of configurations
increasing basis set size
full CI
HF limit
Exact limit