PART I: Essential Computer Experiments PART I ESSENTIAL COMPUTER
EXPERIMENTS
1 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: * int Charge Mult
AtomName-1 NA NB NC
AtomName-2 NA NB NC
...
AtomName-N NA NB NC
*
R A D
R A D
R A D
For example, for H2CO a reasonable input is: * int 0 1
C
0 0 0
O
1 0 0
H
1 2 0
H
1 2 3
*
0.0
1.2
1.1
1.1
000.0
000.0
120.0
120.0
000.0
000.0
000.0
180.0
Building Molecules in the Computer 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° 3 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: 1 Eh = 27.2107 eV = 627.51 kcal/mol = 2625.5 kJ/mol = 219474.2 cm-­‐1 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: 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 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 Interpreting the Results of MO Calculations 5 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): 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. 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 Error! Reference source not found. (page Error! Bookmark not defined.), the orbitals are introduced to the theory as an auxiliary construct. Yet, in Hartree-­‐Fock Interpreting the Results of MO Calculations 6 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. 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. The basic types of molecular orbitals and the principle of their formation from fragment orbitals are shown in Figure 1: σ* π* σ π 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 Interpreting the Results of MO Calculations 7 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: N /2
2
! (r) = 2! "i (r) ( 1) i=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: M
µdip = ! Z AR A " # ! (r) rd r ( 2) A=1
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. Interpreting the Results of MO Calculations 8 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)d r = N ( 3) ( 4) # (r) # (r)d r ! ! (r)d r = " P !!######
"######$
( 5) and from the density matrix P and the basis functions { ! } it follows: ! (r) = ! Pµ"#µ (r)#" (r) µ"
therefore: µ"
µ"
µ
"
Sµ"
= ! 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: AB
= ! ! ! A ! B Pµ!ABS µ!
A
B
µ
!
( 6) Interpreting the Results of MO Calculations 9 AA
AB
= ! ! A ! A Pµ!AAS µ!
+ 2! ! ! A ! B Pµ!ABS µ!
A
µ
!
A B<A
µ
( 7) !
Mulliken proposed to divide the second term equally between each pair of atoms involved and define the number of electrons on center A , N A , as: AA
AB
N A = ! A ! A Pµ!AAS µ!
+ ! ! A ! B Pµ!ABS µ!
µ
!
B"A
µ
( 8) ( 9) !
such that ! N A = N . The charge of an atom in the molecule is then: A
QA = Z A ! N A where Z A 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: AB
BAB = 2! A ! B Pµ!ABS µ!
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. 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 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). Interpreting the Results of MO Calculations 10 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. 3
Table 1: Dipole Moments and Ionization potentials of the small molecules studied in experiment #1. Molecule
CH4
C 2H 6
C 2H 4
C 2H 2
H3COH
H2CO
Dipole Moment (Debye)
0.000
0.000
0.000
0.000
1.700
2.330
3
Ionization Potential (eV)
12.61±0.01
11.56±0.02
10.51±0.015
11.41±0.01
10.84±0.07
10.86
Experimental data from http://srdata.nist.gov/cccbdb/ and http://webbook.nist.gov/chemistry/ Interpreting the Results of MO Calculations HCOOH
CO2
CO
LiH
LiF
NH3
H 2O
HF
Glycine
1.410
0.000
0.112
5.880
6.330
1.470
1.850
1.820
1.095
11 11.31
13.778±0.002
14.0142±0.0003
7.9±0.3
11.3
10.07±0.01
12.6188±0.0009
16.06
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 Geometry Optimization 13 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: 1
(k )
(k )
(k )
(k )
E(R) = E(R ) + (R ! R )f + (R ! R )T H(R ! R ) + """ 2
( ( 11) where the gradient is defined as fi =
!E(R)
!Ri R=R(k)
(12) where R0 refer to the and the Hessian matrix or the force constant matrix is H ij =
13) !E(R)
!R i !Rj
(k )
R=R
Geometry Optimization 14 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. 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. 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: 1
Equad (R) = E(R) + (R ! R)f + (R ! R)T H(R ! R) + """ 2
( 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 Equad (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: Geometry Optimization !Equad
!Ri
= fi + # (Ri " Ri )H ij 15 ( 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"1f ( 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. Geometry Optimization 16 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 rCH in CH4 aHCH in CH4 rCC in C2H6 rCH in C2H6 aHCH in C2H6 aHCC in C2H6 rCC in C2H4 rCH in C2H4 aHCH in C2H4 aHCC in C2H4 rCH in C2H2 rCC in C2H2 aHCC in C2H2 rCC in C6H6 rCH in C6H6 aCCC in C6H6 aHCC in C6H6 rLiH in LiH rLiF in LiF rNH in NH3 aHNH in NH3 aXNH in NH3 rOH in H2O aHOH in H2O rHF in HF Exp. 1.094 109.47 1.536 1.091 108.0 110.91 1.399 1.086 117.6 121.2 1.063 1.203 180.0 1.397 1.084 120.0 120.0 1.596 1.564 1.012 106.67 112.15 0.958 104.48 0.917 Calc. Parameters rOH in H3COH rCO in H3COH rCH in H3COH aHCH in H3COH aHOC in H3COH dHCOH in H3COH rCH in H2CO rCO in H2CO aHCH in H2CO aHCO in H2CO rCO in HCOOH rCH in HCOOH rOH in HCOOH aOCO in HCOOH aHCO in HCOOH aHOC in HCOOH rCO in CO2 aOCO in CO2 rCO in CO rCN in glycine rCC in glycine rCO in glycine rOH in glycine rNH in glycine rCH in glycine aCCN in glycine aCCO in glycine aHOC in glycine aHNC in glycine aHNH in glycine aHCH in glycine Exp. 0.956 1.427 1.096 109.03 108.87 180.0 1.111 1.205 116.133 121.9 1.202, 1.343 1.097 0.972 124.9 124.1 106.3 1.162 180.0 1.128 1.469 1.532 1.207, 1.357 0.974 1.014 1.096 113.0 125.0, 111.5 110.5 113.27 110.29 107.04 Calc. Geometry Optimization 17 Geometry Optimization 18 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 EZPE =
3N !6
" h!
k=1
k
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 ( 18) Geometry Optimization 19 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
!
"i
kT
#e
!
"j
kT
( 19) where N i 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: Geometry Optimization 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. 20 Geometry Optimization 21 Table 3: Relative energies of the isomers of C2H2O2 and C4O6 in kJ/mol. Trans-­‐ Cis-­‐ Cyclo-­‐ C2H2O2 0 C4H6 0 16(5 17(6 46(7 -­‐ 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
7 ENGELN R, CONSALVO D, REUSS J, CHEMICAL PHYSICS 160 (3): 427-433 MAR 15 1992
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