ESSENTIAL COMPUTER EXPERIMENTSscienide2.uwaterloo.ca/~nooijen/Chem-440-computational/... ·...

PART I: Essential Computer Experiments 1

PART I

ESSENTIAL COMPUTER

EXPERIMENTS

Building Molecules in the Computer 2

1 Computer Experiment 1: Building Molecules in the Computer

1.1 Background In this basic and quick experiment you should practice the building of molecular

structures using either paper and pencil or a graphical molecular editor. Preferably you

try both routes.

1.1.1 The Z-‐Matrix If the molecule under investigation is not too large, it may be the most convenient route

to manually input bond distances, bond angles and dihedral angles. A definition of the

molecular structure in this way is called a ‘Z-‐Matrix’. In order to specify the position of a

general atom, one needs six nubers: NA, NB, NC are the numbers of atoms that the new

atom is conected with and R, A and D are a bond distance, a bond angle and a dihedral

angle. The definition of NA, NB and NC is:

• NA: The atom that the actual atom has a distance with

• NB: The actual atom has an angle with atoms NA and NB

• NC: The actual atom has a dihedral angle with atoms NA,NB and NC. This is the

angle between the actual atom and atom NC when looking down the NA-‐NB axis.

Angles are always given in degrees! The first atom is always placed at the origin. The

second atom is placed a distance R (by default in Å units) along one of the coordinate

axes (e.g. X). The third atom is placed in (say) the XZ plane and any further atom really

needs all six specifiers mentioned above.

The format for ORCA is:

For example, for H2CO a reasonable input is:

* int 0 1 C 0 0 0 0.0 000.0 000.0 O 1 0 0 1.2 000.0 000.0 H 1 2 0 1.1 120.0 000.0 H 1 2 3 1.1 120.0 180.0 *

* int Charge Mult AtomName-1 NA NB NC R A D AtomName-2 NA NB NC R A D ... AtomName-N NA NB NC R A D *

Building Molecules in the Computer 3

1.2 Description of the Computer Experiment

1. CH4 : Td symmetry

2. C2H6 : C3v symmetry

3. C2H4 : D2h symmetry

4. C2H2 : D ∞h symmetry

5. H3COH : CS symmetry

6. H2CO : C2v symmetry

7. HCOOH : CS symmetry

8. CO2 : D ∞h symmetry

9. CO : C ∞v symmetry

10. LiH : C ∞v symmetry

11. LiF : C ∞v symmetry

12. NH3 : C3v symmetry

13. H2O : C2v symmetry

14. HF : C ∞v symmetry

15. Glycine : C1 symmetry

You can assume the following “standard” geometrical parameters. C-‐C : 1.54 Å C=C : 1.34 Å C≡C : 1.22 Å C-‐O : 1.40 Å C=O : 1.20 Å C≡O : 1.13 Å C-‐H : 1.10 Å N-‐H : 1.05 Å O-‐H : 1.00 Å Li-‐F : 1.57 Å Li-‐H : 1.62 Å Dihedral angle : 109.4712°

Interpreting the Results of MO Calculations 4

2 Computer Experiment 2: Interpreting the Results of MO

Calculations

2.1 Background In this experiment you will for the first time get subjected to a MO calculation. The goal

of the experiment is to familiarize yourself with the basic quantities that you get from

such calculations.

2.1.1 Total Energy The total energy of a molecule is minus the energy that it takes to separate all particles

(electrons and nuclei) in the molecule and put them in infinite distance of each other.

Thus, the total energy is a very large number. It is measured in Hartree units (also called

“atomic units”) which is abbreviated with the symbol Eh. It is very useful to remember

conversion factors to more chemically relevant units of energy:

Although this is not good scientific practice the majority of the quantum chemical

literature still reports energy differences in kcal/mol (using 1 eV=23.06 kcal/mol and 1

kcal/mol=4.184 kJ/mol). The spectroscopic literature is dominated by the units of cm-‐1

(using 1 eV=8065.73 cm-‐1). It is a very good idea to familiarize yourself with several

units of energy.

For example, the nonrelativistic total energy of the CO molecule is somewhere around -‐

113.X Eh which amounts to ~71000 kcal/mol. Chemically relevant energy differences

are on the order of 1 kJ/mol. This puts the tremendous task of quantum chemistry into

perspective:

Fortunately, we do not need to compute the total energy of molecules to an accuracy of 1

kJ/mol. If this would be the case, quantum chemistry would be a very frustrating

research field. Only in recent years it is possible to reach such a high absolute accuracy

and this is only possible for very small molecules. In chemistry, we are always

measuring energy differences and upon taking these differences most of the errors that

we make in computing the total energies will cancel. In fact, most of the total energy

In quantum chemistry we are facing the challenge of having to compute

small differences between large numbers and to do this with high accuracy.

You have to worry at least about the third digit in the total energy (given in

1 Eh = 27.2107 eV = 627.51 kcal/mol = 2625.5 kJ/mol = 219474.2 cm-‐1


comes from the very strong interaction of the core level electrons with the nuclei and

such core electrons do not contribute appreciably to the chemical behaviour of atoms in

molecules. Still, the basic problem of quantum chemistry is the one of reaching the high

accuracy that is necessary in order to cope with energy differences that are quite small

on the molecular scale but that are dominant for the chemical behaviour of a molecule. It

is relatively easy, for example, to recover ~99% of the total energy – already the

Hartree-‐Fock method is good enough to do that. Yet, the remaining error is huge on the

chemical scale.

2.1.2 Orbital Energies In Hartree-‐Fock theory, each canonical molecular orbital is associated with a unique

molecular orbital (MO) energy. Unlike the total energy, these orbital energies do not

have an “absolute” meaning since they have been introduced into the theory only in

order to satisfy the orthonormality constraint between different molecular orbitals.

Fortunately, the orbital energies of the occupied orbitals can be given an approximate

interpretation (Koopman’s theorem):

This theorem thus makes the important prediction that minus the orbital energy of the

HOMO (the highest occupied MO) is approximately equal to the ionization potential of

the molecule. Furthermore, by plotting the orbital energies as vertical bars on graph that

has ‘orbital energy’ on the x-‐axis, one should obtain a good idea where to expect peaks in

the photoelectron spectrum of the molecule. This is a rather nice connection between MO

calculations and spectroscopy and therefore the canonical orbitals are also called

“spectroscopist’s orbitals”. Upon comparing calculation and reality you will find

important deviations. It is well worthwhile to think about the origin of such

discrepancies!

2.1.3 Molecular Orbitals and Their Shapes Unlike the total Hartree-‐Fock N-‐electron wavefunction and its associated charge density

and despite all claims to the contrary made in chemical textbooks, the orbitals

themselves do not have a rigorous physical meaning. As already discussed in section

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orbitals are introduced to the theory as an auxiliary construct. Yet, in Hartree-‐Fock

The orbital energy of a given canonical MO is approximately equal to minus

the energy that it takes to remove an electron from this orbital. Thus, it is

approximately equal to the first or a higher ionization potential.


theory, each orbital describes the motion of one electron and the square of the orbital

describes its probability distribution. We will come back to the subject of HOMO/LUMO

and popular reactivity arguments in section Error! Reference source not found. (page

Error! Bookmark not defined.). In this computer experiment we only want you to

enjoy looking at orbitals, to define their character as π-‐ or σ-‐orbitals or lone-‐pairs and to

deduce the symmetry labels of these orbitals using group theory.

The basic types of molecular orbitals and the principle of their formation from fragment

orbitals are shown in Figure 1:

Figure 1: Basic types of molecular orbitals.

In the left panel the formation of a homopolar bond is exemplified -‐ two isoenergetic,

singly occupied fragment orbitals form a standard two-‐electron bond. The lower

component is bonding and features constructive overlap of the fragment orbitals; the

Recall that the canonical orbitals transform under the irreducible

representations of the point group that the molecule belongs to. Also

recall that the total symmetry of the state under investigation can be

deduced from the symmetries of the singly occupied orbitals in a given

electronic configuration. Each completely filled subshell is totally

symmetric. Thus, closed shell molecules have a totally symmetric

ground state.

σ* π*

π σ


higher MO is more destabilized than the lower one is stabilized1 and is antibonding. The

formation of a heteropolar bond is shown in the middle panel. Here two orbitals of

different energy interact. The initially higher lying orbital is destabilized and becomes

antibonding. The larger the energy gap and the smaller the orbital interaction, the more

the orbital retains its initial character. Likewise, the lower energy component becomes

bonding but also retains the character of the originally lower-‐lying fragment orbital φb.

The polarity of the bond depends on the energy gap between the two initial fragment

orbitals and their mutual interaction which may be taken to be proportional to the

fragment orbital overlap. The right panel shows some typical members of fragment

orbitals, namely a σ* antibonding MO (usually very high in energy), a π*-‐orbital, a lone-‐

pair orbital as well as a σ-‐bonding and a π-‐bonding orbital. The bond order of a given A-‐

B bond is defined as one-‐half the number of electrons in the bonding orbitals minus the

number of electrons in the antibonding orbitals. The bond order is indicative of but not

directly proportional to the bond dissociation energy which, of course, depends on many

factors.

2.1.4 The total Charge Density, Moments and Population Analysis In Hartree-‐Fock and DFT theory, the total electron density is given as a sum of

contributions of the individual orbitals that make up the single Hartree-‐Fock or Kohn-‐

Sham determinant. For a closed-‐shell system this is:

! r( ) = 2 "

ir( )

2

i=1

N/2

! ( 1)

Where the factor of 2 arises due to the fact that each MO is doubly occupied. From the

total charge density one can computer the various moments of the charge distribution.

The most important is of course the dipole moment and it is related to the polarity of

the molecule. The dipole moment is on observable. It is computed from the charge

density and the nuclear positions RA and nuclear charges ZA as follows:

µ

dip= Z

AR

AA=1

M

! " ! r( )rdr# ( 2)

Where the minus sign arises from the negative charge of the electrons. As it stands the

dipole moment is given in atomic units. In order to convert to the more convention unit

(Debye) one has to multiply the computed dipole moment given in a.u. by 2.541798. The

1 This is seen from the normalization factors involving the fragment overlap integral S.


dipole moment is a vector that points from the center of negative charge of the molecule

to the center of positive charge.

An important concept of chemistry is that of a partial charge of an atom in a molecule.

Unlike the dipole moment the partial charges are not observables. Unfortunately, it

seems to be impossible to arrive at a unique decomposition of the total electron density

(which is a continuous function of space) into parts that “belong” to individual atoms.

Many different attempts have been made to arrive at an approximate decomposition and

these procedures are collectively referred to as “population analysis”. None of these

schemes can claim any rigorous physical reality. Yet, if viewed with appropriate caution,

these schemes can tell you a lot about the trends of the charge distribution in a series of

related molecules. Consequently, almost all quantum chemical programs print one or the

other form of population analysis in their output files. For example, ORCA prints by

default, the Mulliken analysis, the Löwdin analysis and the Mayer analysis. We briefly

review the origin of the Mulliken analysis below:

The Mulliken population analysis is, despite all its known considerable weaknesses, the

standard in most quantum chemical programs. It partitions the total density using the

assignment of basis functions to given atoms in the molecules and the basis function

overlap. If the total charge density is written as !!r( ) and the total number of electrons

is N we have:

! r( )dr! = N ( 3)

and from the density matrix P and the basis functions { ! } it follows:

! r( ) = P

µ"#

µr( )#" r( )

µ"! ( 4)

therefore:

! r( )dr! = Pµ"#

µr( )#" r( )dr!

Sµ"

! "###### $######µ"" ( 5)

= Pµ!S

µ!µ!!

Where S

µ! is the overlap integral between the basis functions µ and ν. After assigning

each basis function to a given center (A,B,C…) this can be rewritten:

= A BPµ!ABS

µ!AB

!!

µ!

B!

A! ( 6)


= A APµ!AAS

µ!AA

!!

µ!

A! + 2 A BP

µ!ABS

µ!AB

!!

µ!

B<A!

A! ( 7)

Mulliken proposed to divide the second term equally between each pair of atoms

involved and define the number of electrons on center A , NA, as:

N

A= A AP

µ!AAS

µ!AA

!!

µ! + A BP

µ!ABS

µ!AB

!!

µ!

B"A! ( 8)

such that

NA

A! = N . The charge of an atom in the molecule is then:

QA= Z

A!N

A ( 9)

where ZA is the core charge of atom A . The cross terms between pairs of basis

functions centered on different atoms is the overlap charge and is used in ORCA to define the Mulliken bond order:

B

AB= 2 A BP

µ!ABS

µ!AB

!!

µ! (

10)

In the present computer experiment you should look at the results of the population

analysis schemes and try to determine whether the observed trends compare well with

your chemical intuition.

TIP:

• A more advanced method of population analysis is the so-‐called “natural

population analysis” invented by Weinhold and co-‐workers. Among the

available choices this one may be recommended for your chemical applications.2

The NPA analysis is available in both ORCA (NPA keyword).

2 A full discussion may be found in F. Weinhold and C. R. Landis, Valency and Bonding: A Natural Bond Orbital Donor-‐Acceptor Perspective (Cambridge U. Press, 2003).

Be careful: In addition to the theoretical problems with population analysis

schemes mentioned above you MUST know that population analysis schemes

are sensitive to the basis set use and do not converge to a well defined basis

set limit. Therefore – when you compare population analysis results for

different molecules: make sure that you have done the calculation with

identical basis sets. Do not compare absolute populations between different


2.2 Description of the Experiment Take the molecules that you made in the first experiment and run a RHF calculation with

the SVP basis set. Look at the following quantities:

1. Look at the results of the population analysis and create a table of partial

charges of, say, the carbon atoms in a series of molecules. How do the

numbers compare with your intuition?

2. Look at the frontier orbitals of the molecules using a visualization package.

Classify the MOs as π, π*, σ, σ* or as lone pair.

3. For at least one of the compounds studied make a quantitative MO scheme.

This should consist of the occupied and the first three unoccupied MOs. Find

the irreducible representations of all MOs and label them on the plot. Are

degenerate MOs unique?

Compare the results of these calculations with the experimental data collected in Table 1

below.

4. Determine the ionization potential predicted by Koopman’s theorem

5. Determine the dipole moment printed at the end of the output.

6. Perform a regression analysis of the computed data using the XMGrace

program. Determine the average absolute error, the largest absolute error, the

average deviation from experiment and the standard deviation. These

quantities are indicative of the reliability of the calculations and the tendency

to over-‐ or underestimate a given quantity.

Table 1: Dipole Moments and Ionization potentials of the small molecules studied in experiment #1.3

Molecule Dipole Moment (Debye) Ionization Potential (eV) CH4 0.000 12.61±0.01 C2H6 0.000 11.56±0.02 C2H4 0.000 10.51±0.015 C2H2 0.000 11.41±0.01 H3COH 1.700 10.84±0.07 H2CO 2.330 10.86

3 Experimental data from http://srdata.nist.gov/cccbdb/ and http://webbook.nist.gov/chemistry/


HCOOH 1.410 11.31 CO2 0.000 13.778±0.002 CO 0.112 14.0142±0.0003 LiH 5.880 7.9±0.3 LiF 6.330 11.3 NH3 1.470 10.07±0.01 H2O 1.850 12.6188±0.0009 HF 1.820 16.06 Glycine 1.095 8.9 The direction of the dipole moments (arrow points from negative to positive)

Geometry Optimization 12

3 Computer Experiment 3: Geometry Optimization

3.1 Background The purpose of this experiment is to locate the most stable arrangement of the

molecules under study. In the case of a diatomic molecule, geometry optimization is

employed to search for the suitable inter-‐atomic distance between these two atoms,

which give rise to the lowest energy among the all conformations of this molecule.

3.1.1 Potential Energy Surface (PES) The way in which the energy of a molecule system varies with the coordinates is usually

referred to as the potential energy surface (PES), sometimes called the “hyper-‐surface”.

Except for the very simplest systems, the PES is a complicated, multidimensional

function of all degrees of freedom of the molecule. For a non-‐linear molecule with N

atoms, the energy is thus a function of 3N-‐6 internal coordinates; it is therefore

impossible to visualize the entire energy surface except for some simple cases where the

energy is a function of just one or two coordinates.

A typical PES is depicted below, each point corresponds to the specific arrangement of

the N atoms in the molecule; hence, each points represents a particular molecular

structure, with the height of the surface at that point corresponding to the energy of that

structure.

Figure 2: Schematic PES adapted from “Exploring Chemistry with Electronic Structure Methods, Second Edition”.

There are three minima on this PES. A minimum is the bottom of a valley on the PES, any

movement away from such a point gives a configuration with a higher energy. A

minimum can be either a local minimum or a global minimum (the lowest energy on the


entire PES). Minima occur at equilibrium structures for the system, with different

minima corresponding to different conformations or structural isomers in the case of

single molecule, or reactant and product molecules in the case of multi-‐component

systems. A point which is a maximum in one direction and a minimum in the all others is

called a saddle point (more precisely a first-‐order saddle point). A saddle point

corresponds to a transition structure connecting the two equilibrium structures, or a

transition state “connecting” the reactant and product.

3.1.2 Searching for Minima Geometry optimizations usually attempt to locate minima on the PES, thus predicting

equilibrium structures of molecular system. Optimizations can also locate transition

states which may be desired or undesired. We will come back to methods for finding

transition states in section Error! Reference source not found. (page Error!

Bookmark not defined.).

At both minima and saddle points, the first derivative of the energy (gradient) with

respect to every internal degree of freedom is zero. Since the gradient is the negative of

the force, it means that at such points the forces are zero as well. Points at which the

gradient of the energy vanishes are called stationary points. They may represent true

minima or saddle points of some kind.

The energy E of a molecular system obtained under the Born-‐Oppenheimer

approximation is a paramertric function of the nuclear coordinates denoted as R, the

energy can be expanded in a Taylor series about the point R(k) as follows:

E(R) = E(R k( ))+ (R!R

k( ))f +12(R!R

k( ))T H(R!Rk( ))+ """ (

11) where the gradient is defined as

fi

=!E(R)!R

i R=Rk( )

(12)

where R0 refer to the and the Hessian matrix or the force constant matrix is

H

ij=!E(R)!R

i!R

j R=Rk( )

(

13)


The energy functions of molecules are hardly quadratic and the Taylor series expand can

only be considered as an approximation, known as harmonic approximation. Close to

minima, it is supposed that a quadratic form is adequate for description of the PES.

For a stationary point R , by definition we require f(R) = 0 , in order to identify this

stationary point is a local minimum other than a saddle point the following condition

must be met:

!i(R) > 0 where !i(R) is the i’th eigenvalue of the Hessian matrix after the

translations and rotations have been projected out.

This corresponds to the condition that there is no imaginary frequency in the frequency

calculation.

Nevertheless for a first-‐order saddle point, the following conditions are necessary:

f(R) = 0 , and

!i(R) < 0 for one specific coordinate (internal reaction coordinate)

!i(R) > 0 for all other coordinates within the molecule. Exactly one imaginary

frequency is indicative of a first-‐order saddle point. In the similar way we can define

higher order saddle point according to the number of imaginary frequency.

3.1.3 Optimization Techniques There are a number of numerical methods for finding stationary point of a function of

many variables. Here a short introduction of widely adopted Newton-‐Raphson (NR)

method is presented below.

Close to a stationary point, a Taylor series expansion of the energy of the molecule under

study is valid:

E

quad(R) = E(R)+ (R!R)f +

12(R!R)T H(R!R)+ """ ( 14)

If R is close enough to R , we are in the quadratic regime it is legitimate to replace the

exact surface E(R) with the quadratic model surface E

quad(R) . It is now straightforward

to minimize the energy of this model surface. The first derivative of the model surface

with respect to a nuclear coordinate is:

Be careful: geometry optimization only searches for stationary points, thus you never

know whether the obtained structures locate at a local minimum or a saddle points. In

order to settle this point it is necessary to perform a frequency calculation on the

optimized structure.


!Equad

!Ri

= fi+ (R

i"R

i)H

ij

j

# (

15)

It is now straightforward to solve for the step-‐vector != R"R which brings us from

point R to the desired point R : in fact:

!="H"1

f (

16)

This equation is the essence of the NR method. Thus, the NR algorithm can locate the

minimum in a single step for a purely quadratic surface. Close enough to the quadratic

regime it is still converging quadratically to the desired stationary point (this means in

practice in very few iterations, e.g. less than five). However, for real surfaces, which are

not quadratic, convergence may be considerably slower. In general, convergence slows

down substantially if the present point R is far from the desired stationary point.

While the fast convergence of the Newton-‐Raphson method close to the minimum is

very attractive, there is an important caveat to its practical use: The calculation of the

Hessian matrix is computationally very demanding for large systems. Thus, essentially

all minimization algorithms try to circumvent the calculation of second derivatives in

each step and only work with the energy E(R) and its first derivative. One possibility

that is followed by the majority of the available programs is the so-‐called quasi-‐Newton

method. In this approach, one starts from a guessed Hessian (or one calculated at a

lower level of theory) and improves on it by using the first derivative information from

various previous iterations.4 If this is done carefully and the starting point of the

optimization was not too bad, convergence can usually be achieved in 10-‐40 iterations

depending on the size and nature of the system. In general, floppy molecules are much

more difficult to optimize. In such molecules low energy rotations around single bonds

may lead to very large geometry changes along very soft modes. All optimization

techniques have difficulties with such situations. It is therefore important to guide the

calculation to the desired minimum and to carefully monitor the progress of a geometry

optimization.

4 The details are of no concern in the present context; we simply note for the interested students that most programs make use of the Broyden-‐Fletcher-‐Goldfarb-‐Shanno (BFGS) algorithm to update the approximate Hessian or its inverse. This is usually a good choice since it helps to retain an initially positive definite Hessian positive definite.


3.2 Description of the Experiment 1. Taking at least five of molecules that you constructured in Experiment 1, run

geometry optimization jobs on them using B3LYP/SVP.

2. Compare your optimized structures with experimental data, and summarize in a

table.

3. Deduce the bond nature as single, double, triple…from the critical bond distances,

and compare the calculated bond orders from the Mayer or Löwdin analysis with the

chemical nature of the bonds. Plot the bond order versus the bond distance for a

given bond type (e.g. the C-‐C bonds in C2H6, C2H4 and C2H2).

4. Perform regression and error analysis as you did in Experiment 2. In order to draw

more definitive conclusions you would certainly need to do more than five

molecules.

Table 2: Experimental geometric parameters of the investigated molecules.

Parameters Exp. Calc. Parameters Exp. Calc. rCH in CH4 1.094 rOH in H3COH 0.956 aHCH in CH4 109.47 rCO in H3COH 1.427 rCC in C2H6 1.536 rCH in H3COH 1.096 rCH in C2H6 1.091 aHCH in H3COH 109.03 aHCH in C2H6 108.0 aHOC in H3COH 108.87 aHCC in C2H6 110.91 dHCOH in H3COH 180.0 rCC in C2H4 1.399 rCH in H2CO 1.111 rCH in C2H4 1.086 rCO in H2CO 1.205 aHCH in C2H4 117.6 aHCH in H2CO 116.133 aHCC in C2H4 121.2 aHCO in H2CO 121.9 rCH in C2H2 1.063 rCO in HCOOH 1.202, 1.343 rCC in C2H2 1.203 rCH in HCOOH 1.097 aHCC in C2H2 180.0 rOH in HCOOH 0.972 rCC in C6H6 1.397 aOCO in HCOOH 124.9 rCH in C6H6 1.084 aHCO in HCOOH 124.1 aCCC in C6H6 120.0 aHOC in HCOOH 106.3 aHCC in C6H6 120.0 rCO in CO2 1.162 rLiH in LiH 1.596 aOCO in CO2 180.0 rLiF in LiF 1.564 rCO in CO 1.128 rNH in NH3 1.012 rCN in glycine 1.469 aHNH in NH3 106.67 rCC in glycine 1.532 aXNH in NH3 112.15 rCO in glycine 1.207, 1.357 rOH in H2O 0.958 rOH in glycine 0.974 aHOH in H2O 104.48 rNH in glycine 1.014 rHF in HF 0.917 rCH in glycine 1.096 aCCN in glycine 113.0 aCCO in glycine 125.0, 111.5 aHOC in glycine 110.5 aHNC in glycine 113.27 aHNH in glycine 110.29 aHCH in glycine 107.04



4 Computer Experiment 4: Relative Energies of Isomers

4.1 Background In this experiment you will conduct a frequency analysis at the stationary points of

the potential energy surface. The goal of this experiment is to get a feeling for how

to locate different minima on a given potential energy surface, to characterize their

nature using frequency calculations and to understand the chemical implications of

the different minima.

The necessary theoretical background is collected in section 3 (nature of stationary

points) and section Error! Reference source not found. (meaning of vibrational

and thermal corrections to the total energy). Briefly, the total energy of a molecule consists to a good approximation of additive contributions

from its electronic energy (together with the nuclear repulsion), its translational energy, its

rotational energy and its vibrational energy. The latter contribution may be divided into a part

corresponding to the zero-‐point energy ( Ezpe sum of the energies of all ν=0 levels) and a

thermal correction ( Evib* ) coming from Boltzmann-‐population of the higher vibrational levels of

the system.

Etot = Eele + Etra + Erot + Evib* + Ezpe

( 17) Contributions from translational, rotational and excited vibrational states( Etra , Erot

and Evib* accordingly) are frequently negligible in comparing the energies of

different isomers but the zero-‐point correction may be important. It is obtained

from

E

ZPE= h!

kk=1

3N!6

" ( 18)

With ! k being the k’th vibrational frequency of the molecule.

As you have determined several stationary points and their character on the

potential energy surface which correspond to different conformers or electronic


states, you may be interested in the population of these states at a certain

temperature. Therefore, Boltzmann-‐statistics is employed. In Boltzmann statistics

the fractional population of the i’th state is given by:

Ni = N e!" ikT

e!" j

kT#

( 19)

where Ni is the number of particles in the i’th energy level ! i , N the number of all

particles, k is the Boltzmann-‐constant, T the temperature in Kelvin and the sum

includes all energy states.

4.2 Description of the Experiment Similarly, as in the last experiment, build the Z-‐matrices for the two geometric

confomers of glyoxal (trans-‐, cis-‐) as well as for the three different

confomers/isomers of butadiene (trans-‐, cis-‐) and cyclo-‐butene. Run a B3LYP/DFT

calculation with the SVP basis set.

Figure 3: The two isomers of glyoxal: trans (left) and cis (right).

Figure 4: Three isomers of C4H6: trans (left) and cis (Middle) butadiene and cyclobutene (right).

Conduct the following steps:


1. Execute a full geometry optimization for all conformers and determine the

stationary points on the potential energy surface. Try several starting

geometries (distort the molecule)

Determine whether the obtained stationary points are local minima. To this

end, perform frequency calculations. Use your chemical intuition in order to

guess a starting geometry that leads to convergence to a first-‐order saddle

point. Confirm your suspicion by a frequency analysis.

2. If you have been successful in finding different local minima, compare the

relative energies of each isomer.


Table 3: Relative energies of the isomers of C2H2O2 and C4O6 in kJ/mol.

3. Does inclusion of Zero-‐Point-‐Energy improve the relative energies

significantly? Compare the magnitude of the thermal correction to that of the

ZPE corrections. Which contribution is more significant for relative isomer

energies ?

4. Calculate the fractional population of each isomeric form using Boltzmann

statistics? Will you necessarily observe the different isomers in this

proportion in actual experiments? Discuss possible sources of deviations

from the expected ratios.

5 BUTZ KW, KRAJNOVICH DJ, PARMENTER CS, JOURNAL OF CHEMICAL PHYSICS 93 (3): 1557-‐1567 AUG 1 1990 6 ENGELN R, CONSALVO D, REUSS J, CHEMICAL PHYSICS 160 (3): 427-433 MAR 15 1992 7 SPELLMEYER DC, HOUK KN, J. AM.CHEM.SOC. 110, 11, 3412-‐3416, 1988; WIBERG KB, FENOGLIO RA, J. AM.CHEM.SOC. 90, 13, 3395-‐3397, 1968

C2H2O2 C4H6

Trans-‐ 0 0 Cis-‐ 16(5 17(6 Cyclo-‐ -‐ 46(7

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