Quantum Chemistry
7 Section 6: Methods and Basis Sets (How to work with a computational chemistry program)
8
9 In this section we will explore a number of methods and basis sets, and explore the impact of
10 expanding the method and basis set on calculated properties. The key concept we will explore is
11 convergence of calculated properties. This section forms the foundation of subsequent sections,
12 whereby we can explore the question of how accurate are my results, and how can I assess
13 that?
14
15 We will explore the effect of method and basis set on (i) geometry and (ii) energy of reaction to
16 illustrate how methods and basis sets effect the calculated result.
17
18 Methods and Basis Sets
19
20 The use of a theoretical model (model chemistry) combines a method with an atomic orbital basis
21 set. The method and basis set can be described as follows.
22 • method describes how electrons in atomic orbitals interact with each other (N-electron),
23 • basis set describes the set of atomic orbitals (1-electron).
24 There is a host of abbreviations and notation that are used in computational chemistry. All
25 acronyms have a specific meaning, however in this course we are focused on introducing the
26 concepts and modelling chemistry.
27
28 We generally use the notation of Pople, which is:
29 method/basis set
30 HF/6-31G Hartree-Fock wave function with a 6-31G basis set
31 MP2/cc-pVTZ MP2 wave function with a cc-pVTZ basis set
32 B3LYP/6-31G* B3LYP-DFT method with a 6-31G* basis set
33 CCSD(T)/TZVP CCSD(T) wave function method with a TZVP basis set
34
The two-dimensional chart of nonrelativistic quantum chemistry
• The quality of nonrelativistic molecular electronic-structure calculations is
determined by the description of
1. the N-electron space (wave-function model),
2. the one-electron space (basis set).
• In each space, there is a hierarchy of levels of increasing complexity:
1. the N-electron hierarchy:
coupled-cluster excitation levels
HF, CCSD, CCSDT, CCSDTQ, …
2. the one-electron hierarchy:
correlation-consistent basis sets
DZ, TZ, QZ, 5Z, 6Z, … one!electron basis sets wave!function models
exact solution
• The quality is systematically improved upon by going up in the hierarchies.
6
S6.2
1 Methods
2
3 If solutions are generated without reference to experimental data, but only rely on fundamental
4 constants, they may be called ab initio (latin for ‘from the beginning’). Electronic structure methods
5 are characterized by their various mathematical approximations to the solution of this equation.
6
7 There are three major classes of electronic structure methods:
8 • Semi-empirical methods, such as AM1, PM3, MNDO.
9 • Ab initio methods (Hartree-Fock, Møller-Plesset, coupled cluster, etc)
10 • Density functional theory (DFT) methods (e.g. B3LYP, BP86, etc)
11
12 The Density Functional Theory (DFT) methods have become the most popular methods used in
13 computational chemistry. DFT methods have about the same computational expense of the
14 Hartree-Fock method, but they include electron correlation – the fact that the motion of electrons is
15 affected by other electrons. The Hartree-Fock model only considers this in an average way.
16
17 There are a very large number of methods (and acronyms) in computational chemistry.
18
19 Basis Sets
20
21 A basis set is the mathematical description of the atomic orbitals (AO) of a system, which in turn
22 combine to form molecular orbitals (MOs). Larger basis sets more accurately represent the orbitals
23 by imposing fewer restrictions on the locations of electrons in space (at increasing computational
24 cost).
25
26 Molecular orbitals are constructed as a linear combination of atomic orbitals (basis functions). The
27 original formulation of the Hartree-Fock equations was in terms of Slater type orbitals, however
28 almost all quantum chemistry computer programs utilize Gaussian functions, which are easier for
29 computers to work with. Gaussian functions have the form,
30
STO
€
g(ζ,r) = cx n
y mz
l
e−aζr
€
g(ζ,r) ∝e−ζr 31
GTO
€
g(ζ,r)=cx n
y mz
l
e−aζr 2
€
g(ζ,r) ∝e−ζr 2
32
33
34 where r is composed of x, y and z Cartesian components (angular component). ζ (zeta) is a
35 constant determining the size (radial extent) of the function. A small value of zeta refers to a diffuse
36 basis function that allow the electrons to be further from the nucleus, while basis functions with a
37 large value of zeta are tight functions, in that they are close to the nucleus.
38
39 Basis set families
40
41 There are a few common families of basis sets, including Pople-style basis sets and Dunning’s
42 correlation consistent basis sets (notation cc-pVXZ, X=D,T,Q,5,…).
43
44 A minimal basis set contains the minimum number of basis functions needed for each atom. For
45 example,
S6.3
1 H: 1s
2 C: 1s, 2s, 2px, 2py, 2pz
3
4 The STO-3G basis set is a minimal basis set.
5
6 The first way that a basis set can be made larger is to increase the number of basis functions per
7 atom. Double zeta basis sets include 3-21G, 6-31G and cc-pVDZ. For example, the 3-21G basis
8 includes the functions:
9
10 H: 1s, 1s´
11 C: 1s, 2s, 2s´, 2px, 2px´, 2py, 2py´, 2pz, 2pz´
12
13 where the primed and unprimed orbitals can differ in size. The double-zeta terminology comes
14 from the fact that two Gaussian functions (and hence two values of zeta) are given for each atomic
15 orbital.
16
17 A triple split valence (triple-zeta) includes three basis functions for each valence orbital. The 6-
18 311G and cc-pVTZ basis sets are examples of triple-zeta basis sets.
19
20 Split-valence basis sets employ an approximation (for computational efficiency), by including
21 single-zeta (minimal) description of core orbitals, and the double-zeta (etc) only applies to valence
22 orbitals. Most basis sets are of this form, as most chemistry is dictated by valence rather than core
23 orbitals.
24
25 Polarisation functions are functions of higher angular momentum – they allow for changes in
26 shape. Split-valence basis sets allow orbitals to change size (radial extent), but not shape. That is,
27 a split-valence basis set for H only contains fully symmetric s type orbitals, from which a linear
28 combination will still be totally symmetric. Polarised basis sets remove this restriction by adding
29 basis functions of higher angular momentum. For example, adding p functions to H, d functions to
30 C and f functions to transition metals. A common polarized basis set is the 6-31G(d), which
31 denotes that d functions are added to heavy (non-hydrogen) atoms. This basis set is also known
32 as 6-31G*. We can additionally add polarization functions to the H, which is denoted 6-31G(d,p) or
33 6-31G**.
34
35 Diffuse functions are larger-size s,p,d,… orbitals that allow orbitals to occupy a larger region of
36 space. Basis sets with diffuse functions are especially important for molecules with electrons far
37 from the nucleus, such as anions, molecules with lone pairs, molecules with significant negative
38 charge and systems in excited states. Diffuse functions are added to Pople-style basis sets with
39 the notation 6-31+G(d), which adds diffuse functions to the 6-31G(d) basis set. The 6-31++G(d)
40 adds diffuse functions to both hydrogen and heavy (non hydrogen) atoms. Adding diffuse functions
41 to hydrogens generally has little effect on the accuracy of results. The cc-pVXZ basis sets are
42 extended with diffuse functions by adding an ‘aug-‘ prefix, eg. aug-cc-pVDZ adds a diffuse function
43 for each angular momentum.
44
S6.4
1 The correlation-consistent basis sets are designed to systematically recover the correlation energy
2 (and converge to the theoretical basis set limit). These work very well with CCSD(T) theory. For
3 example, running cc-pVDZ (double-zeta) then cc-pVTZ (triple-zeta) then cc-pVQZ (quadruple-zeta)
4 calculations would give a clear indication of the convergence of the calculated property.
5
6 Basis set convergence: The following plot (left) illustrates the convergence of the Hartree-Fock
7 energy of water as the basis set is expanded. This trend is typical of most molecular properties,
8 including energies, geometries (bond distances, angles), vibrational spectra, NMR shifts etc.
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27 To determine whether a basis set will achieve satisfactory results we can look at the basis set
28 convergence. We will perform the calculations with different basis sets. When enlargement of the
29 basis set hardly influences the outcome, there is ‘convergence of the basis set’, and we can
30 conclude that the quality of the basis set is sufficient.
31
32 Method and Basis Set Convergence 1: Geometry
33
34 In this section we will illustrate the effect of basis set convergence (towards the basis set limit), as
35 well as the effect of diffuse and polarization functions.
36
37 We will study basis set effects on the optimized geometry of the HF molecule. The experimental
38 value is 0.917 Å.
39
40 Perform the following calculations (geometry optimizations) at the Hartree-Fock level of theory.
41 Which (if any) basis sets yield results within 0.01 Å of the experimental value?
42
energy. The Hartree–Fock model allows for an exact treatment
within a finite one-electron basis with
5 orbitals suffi-
cient for H2O⇥, whereas the correlation energy cannot be
treated exactly in any finite-dimensional basis. Consequently,
it is reasonable to expect the convergence characteristics
of these energy contributions to be different. The dominance
of the Hartree–Fock energy over the correlation
energy also makes a separate treatment of these contributions
desirable.
In order to extrapolate the Hartree–Fock and correlation
energies, we assume that simple analytical forms can be
found that display the correct asymptotic behavior of the
energies, and which allow for accurate fits of the calculated
energies with only a few parameters since
only a few energies
can be calculated⇥. Dunning found that, in correlated
calculations, the energy lowerings along the cc-pVXZ sequence
decrease approximately geometrically. Indeed, this
observation constitutes much of the justification for the
correlation-consistent basis sets. Although the energy corrections
in Hartree–Fock calculations are expected to be smaller
than those obtained in correlated calculations, we may still at
this uncorrelated level assume a geometrical progression for
these basis sets. In Fig. 1, we have therefore plotted the
cc-pVXZ energies for the water molecule together with an
exponential fit of the form
ESCF⇤a⇥b expcX⇥,
4⇥
as suggested by Feller.6 This fit is seen to be an excellent one
with errors of the order of 0.1 mEh for all basis sets, indicating
that the error in the SCF energy indeed decreases in a
geometrical fashion for the correlation-consistent basis sets.
For the cc-pVXZ sets, the SCF error decreases by a factor
of 3.9 with each increment in the cardinal number. A
similar exercise for the core-valence sets cc-pCVXZ gives a
plot indistinguishable from those of the valence set and also
a similar fit, with basis-set limit of 76.067
6 Eh and a decrease
by a factor of 3.9 with each increment in the cardinal
number. As expected, there is nothing to be gained in the
total energy by carrying out Hartree–Fock calculations in the
cc-pCVXZ basis set, although molecular properties that depend
critically on the electron distribution in the inner valence
and core regions may be significantly affected.
E. Correlation energies and extrapolations
Since the exponential fit was so successful for the
Hartree–Fock energies and since, for the correlation energies,
a geometrical progression has been observed by Dunning
and co-workers, it would appear natural to use the same
exponential fits for the correlation energies. Indeed, it appears
that reasonable exponential fits can be made to the
calculated correlation energies for the valence and corevalence
calculations. At this point, we recall that there is no
rigorous theoretical justification for the use of the exponential
function for the extrapolation of the correlation-energy
limit. On the contrary, the work of Schwarz indicates that the
correlation energy should not converge exponentially, but as
an inverse power in the highest angular-momentum function
present in the basis set.7 Investigating the second-order energy
of the 1/Z perturbation expansion for two-electron atoms,
Schwartz showed that, asymptotically, the convergence
with respect to the angular-momentum quantum number l
contained in the one-particle basis goes as
⇤El
2⇥
⇤
45
256 ⌅4
1
5
4 ⌅2⇥O⌅4⇥⇥
, 5⇥
where ⌅⇤(l⇥ 1
2)1.
A similar formula was found empirically
by Carroll et al.24 for configuration interaction calculations
on the helium atom,
⇤El⇤0.074⌅40.031⌅5⇥O⌅6⇥,
6⇥
later analyzed in detail by Hill.25 Recently, atomic states
other than the 1S ground state have been studied by Kutzelnigg
and Morgan,8 who also considered the rates of convergence
of n-electron atoms in second-order Mo”ller–Plesset
MP2⇥
perturbation-theory calculations.
Schwarz’ work was based on a 1/Z perturbation analysis
of the wave function of the helium atom and applies to a
situation where the basis sets at each level are completely
saturated with respect to the radial part of the basis functions.
We cannot apply these results directly to the cc-pVXZ and
cc-pCVXZ calculations on the water molecule since these
basis sets are constructed in a ‘‘correlation-consistent’’
manner—that is, convergence is obtained by adding functions
of different angular momenta simultaneously if they
give similar energy lowerings, rather than by saturating each
angular-momentum space separately. Nevertheless, by analogy
with the helium expansion, it is reasonable to expect that
FIG. 1. Convergence of the SCF energy in
Eh⇥ for H2O as a function of the
cardinal number X⇤2,3,…,6 in the series of basis sets cc-pVXZ. The solid
line corresponds to the fit ESCF⇤76.067
6⇥0.617 161 exp
(1.357
71X).
9642 Helgaker et al.: Correlated calculations on water
J. Chem. Phys., Vol. 106, No. 23, 15 June 1997
Downloaded¬06¬Jun¬2003¬to¬129.242.5.30.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcr.jsp
The two-dimensional chart of nonrelativistic quantum chemistry
• The quality of nonrelativistic molecular electronic-structure calculations is
determined by the description of
1. the N-electron space (wave-function model),
2. the one-electron space (basis set).
• In each space, there is a hierarchy of levels of increasing complexity:
1. the N-electron hierarchy:
coupled-cluster excitation levels
HF, CCSD, CCSDT, CCSDTQ, …
2. the one-electron hierarchy:
correlation-consistent basis sets
DZ, TZ, QZ, 5Z, 6Z, … one!electron basis sets wave
!function models
exact solution
• The quality is systematically improved upon by going up in the hierarchies.
6
S6.5
1 Table: Calculated H-F bond distances.
2
Basis set Bond distance (Å)
Hartree-Fock MP2 CCSD(T)
1. Pople Style Basis Sets
6-31G
6-31G(d)
6-31G(d,p)
6-31+G(d,p)
6-31++G(d,p)
6-311G(d,p) 0.8960
6-311++G(d,p) 0.8976
6-311G(3df,3pd) 0.8957
6-311++G(3df,3pd) 0.8972
2. Correlation-consistent basis sets
cc-pVDZ
cc-pVTZ
cc-pVQZ
3. Correlation-consistent basis sets with aug (diffuse) functions
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVQZ
3
4 In this last part of the table we are using the augmented basis sets (have additional diffuse
5 functions) to investigate the effect of including these basis functions. This is important to consider
6 since we have a very electronegative element present (F), and so we need to account for electron
7 density further from the nucleus to produce accurate results.
8
9 Method and Basis Set Convergence 2: Energy
10
11 In this exercise we will calculate the energies of isomers of C2H2F2. There are 3 isomers we will
12 consider: cis-1,2-difluoroethene, trans-1,2-difluoroethene, and 1,1-difluoroethene.
13
14 Create cis-1,2-difluoroethene in WebMO. Optimize the geometry at the BP86/6-311+G(d,p) level of
15 theory.
16
17 Using the optimized geometry, calculate the energy with BP86 and the def2svp basis set an copy
18 the result into the following table. Repeat the calculation with the def2tzvp and def2qzvp basis
19 sets. You will see that the energy is successively lower (more negative).
20
21 As part of the exercise you will calculate the energy of all 3 isomers. We will use the BP86 DFT
22 method together with the following basis sets: def2svp (DZ basis set), def2tzvp (TZ), and def2qzvp
23 (QZ).
S6.6
1 Table: Energy of difluoroethene isomers
2
Cis-1,2 Trans-1,2 1,1
Total energy (Hartree)
BP86/def2svp
BP86/def2tzvp
BP86/def2qzvp
Relative energy (kJ/mol)
BP86/def2svp
BP86/def2tzvp
BP86/def2qzvp
3
4
5 Calculations and Units (a reminder)
6
7 The energy in atomic units (Hartrees) should be retained with ALL significant figures. Because you
8 will convert from a.u. to kJ/mol by multiplying by 2625.50, then at the very least, 6 decimal places
9 (in a.u.) are important!
10
11 It is recommended to leave your energies in atomic units (Hartree), perform any calculations in
12 atomic units (Hartree), and only convert your final answer to kJ/mol. This reduces any round-off
13 errors due to the ‘approximate’ conversion factor (it is more accurately 2625.4997).
14
15 For example,
16 reactants -75.9628659848 Hartree
17 products -76.0177431627 Hartree
18 Difference 0.0548771779 Hartree
19 Difference 144.08 kJ/mol
20
21 In quoting the final answer you should be mindful of significant figures. Throughout the calculation,
22 retain all digits and only round your final answer.
23
24
25
S6.7
1
2
3 Exercise 6: Basis set and Methods
4
5 Learning Outcomes
6 • Explore the effect of basis sets on calculated properties and energies, including the effect of
7 adding diffuse and polarization functions.
8 • Develop an understanding of the concept of basis set convergence (how energies and
9 properties converge to a value as the basis set is expanded), by plotting bond distances and
10 energies as a function of basis set.
11 • Develop an appreciation of the relationship between computational accuracy and computational
12 expense (time of calculation, computer memory/disk required)
13 • Identify the lowest energy isomer of difluoroethene, and rationalize why the trans isomer is not
14 the lowest energy isomer (investigate the so-called cis-effect).
15
16 (a) Geometry optimizations
17
18 (i) Complete the above table for the geometry optimization of the HF molecule, with the HF, MP2
19 and CCSD(T) methods. You will probably have to manually enter the basis sets (select other).
20
21 NOTES:
22 * To run the larger calculations with MP2 and CCSD(T), you will need to increase the memory
23 allocated for the calculation.
24 (i) set up calculation in Gaussian job options (eg, CCSD(T) with aug-cc-pVTZ basis set),
25 (ii) select the preview tab, click on generate (this generates the input file).
26 (iii) Add a single line at the top (above all the text that is there),
27 %mem=400MB
28 This increases the requested memory (RAM) to 400 MB.
29 If the job still fails, increase the memory to 800MB.
30
31 * The longest calculation will take about 10 minutes.
32
33 * You will need to look at the raw output to get the geometry to 4 decimal places (towards the
34 bottom of the file).
35
36 (ii) Pople-style basis sets: Comment on the specific effect on bond distance of adding diffuse
37 and/or polarization functions. Are the trends consistent for all methods?
38
39 (iii) Correlation-consistent basis sets: Comment on the specific effect on bond distances of
40 increasing the size of the basis set (from DZ to TZ and QZ). Consider the cc-pVXZ and aug-cc-
41 pVXZ basis sets separately.
42
43 (iv) Correlation-consistent basis sets: Consider the effect of diffuse functions on calculated bond
44 distance by comparing the results with cc-pVDZ and aug-cc-pVDZ basis sets, and then TZ and
S6.8
1 QZ. That is, how do the aug-cc-pVXZ results compare to those with cc-pVXZ, and is the trend
2 consistent with the results of adding diffuse functions to Pople-style basis sets.
3
4 (v) Correlation-consistent basis sets: Plot your results (optimized H-F bond distance) using the
5 correlation-consistent basis sets (plot cc-pVXZ and aug-cc-pVXZ in separate plots) for each
6 method to observe the basis-set convergence for the optimized bond distance. Put the H-F bond
7 distance on the y-axis, and the basis set (D,T,Q for cc-pVXZ and aug-cc-pVXZ) on the x-axis, with
8 a line for each method considered.
9
10 (vi) Comment on the convergence properties and relative accuracy of the various methods.
11
12 (vi) The Møller-Plesset and Coupled-Cluster (CCSD(T)) methods include electron correlation.
13 Based on your results and analysis (comparing calculated and experimental bond distances), in
14 your opinion, is it necessary to account for electron correlation in calculating an equilibrium
15 geometry of HF?
16
17 (b) Relative energies of C2F2H2 isomers.
18
19 Construct cis-1,2-difluoruethene, trans-1,2-difluoroethene, and 1,1-difluoroethene. Optimize the
20 geometry of each isomer at the BP86/6-311+G(d,p) level of theory.
21
22 Calculate the relative energies of the 3 isomers using the def2svp, def2tzvp, and def2qzvp basis
23 sets and the following methods:
24 (i) Hartree-Fock
25 (ii) BP86
26
27 Put your results into a table.
28
29 You can find highly accurate values for these relative energies from a search of the Internet – try
30 Google as a starting point. Include in your report a discussion that compares your calculated
31 values with an accurate reported result. Discuss your results, including an analysis of basis set
32 convergence
P(5.u)
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