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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