Transcript 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 Semi-empirical methods also available Variational methods Perturbative methods also available
not
mean exact.
The Variational Principle E
0
(exact) < E
0
(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)
A
How to describe a molecule
R r iA r i e i B r iB r i defines position of electron
i
s i is the spin coordinate of electron
i
x
indicates centre of mass of system Separating total translation of molecule:
H
1 2 2
R
i N
1 1 2 2
r i
i N
1
Z A r iA
i N
1
Z B r iB
i N N
1
j i
1
r ij
Z A Z B R
H el
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
i N
1 1 2 2
r i
i N
1
Z A r iA
i N
1
Z B r iB
i N N
1
i
1
r ij
How to describe electronic states
H el
i N
1 1 2 2
r i
i N
1
Z A r iA
i N
1
Z B r iB
i N N
1
i
1
r ij
• Can’t be solved exactly (except for H 2 + ) • Must use approximate methods • Implementations make use of molecular symmetry to simplify numerical calculations We could write the multielectronic wavefunction as a product of 1 particle functions: MOLECULAR METHOD = molecular orbital = f(x,y,z) Spin-orbital = x spin function = f(x,y,z,s,s z ) 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 (
r
1 ) 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 , B 3g ) 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 (
r
1 ) 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 N th electron can be approximated as an averaged field Hartree-Fock equation:
F
i
i
i F
h
j N
1 ( 2
J j
K j
)
J j
and
K j
are the coulomb and exchange operators and they all the other orbitals.
depend on *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
e
r
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) Contracted GTO:
i L
1
C i
(
i
) C i 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
i
(
r
)
M
j a ij
(
r
)
a ij
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. H 2 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 (
r
1 ) 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’ E corr =E exact - E HF-limit
full CI increasing basis set size HF limit