PART II ADVANCED AND SPECIAL SUBJECTS

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1 PART II ADVANCED AND SPECIAL SUBJECTS 1 PART II ADVANCED AND SPECIAL SUBJECTS

2 PART II ADVANCED AND SPECIAL SUBJECTS 2 1 Computer Experiment 7: Interpretation of Structure, Bonding and Reactivity Using Orbitals 1.1 Background Hückel MO Theory The Hückel theory is the simplest semi- empirical method for the description of molecular systems. It was developed in 1931 by Erich Hückel for the calculation of planar conjugated hydrocarbons. Later Hoffmann generalized this very successful concept to general molecules and applied it with great success to many areas of chemistry. He termed his method the extended Hückel theory (EHT). The appeal of the Hückel and EHT methods is that they can provide many qualitative insights into the behaviour of molecules, in particular of molecules that are closely related. 1 It does not, however, provide reliable numbers. For this purpose ab initio or DFT methods must be consulted. 2 A. Extended Hückel MO theory EHT theory might be thought of as an essentially very simple, semiempirical approximation of Hartree- Fock theory. In Hartree- Fock theory, a single Slater determinant is taken as an Ansatz for the description of the N- electron system. The single determinant is composed of one- particle molecular orbitals (MOs) that are found as solutions to a pseudo- eigenvalue problem of the form: ˆF! i = " i! i ( 1) Where ˆF is the Fock operator, which describes the motion of a single electron in the field of the nuclei and the remaining electrons.the MOs are written as a linear combination of atomic orbitals (LCAO- Ansatz): 1 Recommended literature to this chapter is: Ian Fleming, Grenzorbitale und Reaktionen organischer Verbindungen, VCH Even today it might not be a bad idea to do a few simple EHT calculations at the beginning of a project if one has no familiarity with the systems being investigated and to use the results to gain some feeling for the factors that might be worthwhile to examine with more rigorous electronic structure methods.

3 PART II ADVANCED AND SPECIAL SUBJECTS 3 ( ) =! c µi " µ ( r)! i r µ ( 2) After which the calculation boils down to the solution of a generalized eigenvalue problem for the determination of the unknown MO coefficients c µi and the orbital energies! i Fc =!Sc ( 3) Where! is a diagonal matrix with orbital energies and the matrix elements of the Fock matrix F and the overlap matrix S are defined by: F µ! = " µ ˆF "! ( 4) S µ! = " µ "! ( 5) As pointed out in the introduction, Hartree- Fock calculations require the use of large basis sets and iterative cycles for the optimization of the MOs. In EHT theory, one focuses on the qualitative shape of the valence orbitals and usually pays attention to those orbitals that are located near the HOMO- LUMO gap (the frontier orbitals ). According to frontier molecular orbital theory (described below in chapter 1.1.3) these are the most important ones for the reactivity of the system. In order to get an impression of how these orbitals may look like it is not necessary to solve the rather laborious HF equations in fact this was a major challenge for all but the smallest molecules in the 1960s when EHT theory was suggested it is enough to replace the Fock operator by a semiempirical effective one- electron operator ĥeff and to only consider the minimal chemical set of valence orbitals 3 as {!}. It is not necessary to specify the operator ĥeff precisely. In order to solve eq (3), it is only necessary to specify the matrix elements of this operator. In EHT theory they are given by: 3 For example, for a carbon atom one includes 2s, 2p x, 2p y and 2p z and for titanium one would use 3d xy, 3d x2- y2, 3d xz, 3d yz, 2d z2- r2, 4s and perhaps also 4p x, 4po y and 4p z.

4 PART II ADVANCED AND SPECIAL SUBJECTS 4 eff h µ! " $ = " µ ĥ eff "! = # if µ =! µ # $ $ µ! if µ!! % $ ( 6) The diagonal elements! µ represent the energy of the atomic orbital! µ in the free atom. It is approximated by the so- called valence shell ionization energy (VOIP) which can be determined from atomic spectroscopy. 4 These numbers are negative and represent the average energy required to remove an electron from the given orbital (2s, 2p, 3d, ) The lower these energies, the higher the electron attracting ability of the atom. The off- diagonal matrix elements are called resonance integrals. They measure the strength of the interaction between two atomic orbitals. Intuitively, it is reasonable to expect that these integrals are related to the overlap integrals. Thus, in EHT they are given by: 5 ( )! µ" = 1 2 KS µ" # µ + # " ( 7) The constant K ~ 1.75 is an empirical constant to adjust the resonance integrals to more reasonable values given that the form of eq (7) is an oversimplification. In order to provide a feel for the values of the parameters we give them for elements H- Ne in Table 1:. Table 1: VOIPs for elements H- Ne to be used together with EHT calculations. Note, that the negative of the values listed below is to be used. Element αs (ev) αp (ev) H He Li Be B C N The values of these parameters are tabulated in multiple places. Since there are many ways to determine these parameters from the experimental data, no consensus has been reached in the community about a universal set of VOIPs. 5 There are many modifications of the formulas for the off- diagonal. It is questionable if any of these is really to be preferred over another one and in this situation Ockham s razor principle applies which essentially states that the simplest solution is just as good as any.

5 PART II ADVANCED AND SPECIAL SUBJECTS 5 O F Ne Following the solution of the generalized eigenvalue problem in eq (3) and with the approximation described above, the resulting MOs are filled in order of increasing energy with the available electrons in keeping with the Aufbau principle in order to find the electronic ground state. Since the approximations are so simple, the total energy in the EHT method is simply the sum of orbital energies: E EHT = " n i! i ( 8) i Where n i is the occupation number of the i th orbital (=0,1 or 2). In connection with Walsh s rules, we will see below that the variations of the orbital energies with geometry can be used to obtain quite important and general insights about the geometric structures of molecules. In fact, many areas of chemistry have profited from performing such qualitative calculations. B. Hückel theory for π electron systems (HMO) Although it existed prior to EHT theory, the HMO theory of aromatic π- systems might be thought of as a further simplification of the EHT method. In this case, one only considers the π- electrons of the investigated molecules and only keeps the pz orbitals of the atoms involved in the π- system. Substituent effects are included in an approximate manner by modifying the α- parameters of the atoms to which the substituents are attached. Furthermore, in HMO theory, the off- diagonal elements of the overlap matrix are neglected such that the final eigenvalue problem to be solved is: h eff c =!c ( 9) Finally, one only keeps resonance integrals between atoms that are nearest neighbours and assigns a constant value to them. Thus, the calculations also become geometry independent and the only thing that is required in order to perform a HMO calculation is a molecular connectivity. Although these approximations clearly

6 PART II ADVANCED AND SPECIAL SUBJECTS 6 represent a gross oversimplification of the molecular electronic structure problem, it can hardly be overemphasized how much HMO theory has shaped chemical thinking about aromatic molecules and their properties. 6 However, we will not pursue HMO theory in this course but stay in the framework of the EHT method below Walsh s Rules 7 Walsh diagrams are the graphical representation of the MO energies in dependence of a geometrical parameter, most commonly a bond angle. For example, consider the case of the H3 molecule in its linear and triangular forms as shown in Figure 1. It is observed that upon bending, the energy of the nonbonding σu orbital (it correlates with a b2 orbital in C2v symmetry) strongly increases and finally correlates with one component of the antibonding e- orbital in the final D3h structure. Since there are three electrons to be filled into the three MOs, it is predicted that H3 should be linear ((1σg) 2 (1σu) 1 configuration) while H3 + should be triangular ((a1) 2 configuration) since the a1 orbital is stabilized upon bending. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Figure 1: Walsh diagram for the distortion of the H 3 system. The variation of a geometrical degree of freedom may cause a crossing of different MO- levels. In general, such a crossing is allowed if the two MOs transform under different irreducible representations and is avoided otherwise.(the famous non- crossing rule; compare Figure 2). The non- crossing rule is particularly important for the interpretation of photochemical reactions. The occurrence of a HOMO / LUMO 6 The classic text on HMO theory is E. Heilbronner, H. Bock Das HMO Modell und seine Anwendung. Verlag Chemie, Weinheim/Bergstrasse, For a detailed ab initio perspective on Walsh s rules see: Buenker, R.J.M; Peyerimhoff, S.D. Chem. Rev., 1974, 74, 127

7 PART II ADVANCED AND SPECIAL SUBJECTS 7 crossing with identical symmetry implies that the reaction is symmetry forbidden, due to the non- conservation of the orbital symmetry of the occupied MOs. Therefore a high activation energy is expected. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Figure 2: Crossing and avoided crossing of two energy levels. In Figure 3 the Walsh diagram for the bending mode of a AH2 molecule is shown. The energy of the lowest valence MOs decrease upon bending as three center bonding between s- AOs becomes more favourable in the bent form. The 1 σu- MO is strongly bonding in the linear case, but upon reducing the angle, the 1b2- MO becomes only weakly bonding, because in the linear case the 2py- AOs can interact more strongly with the AOs of the H- atoms. The 1 πu- MO separates in two components, 1b1 and 3a1. The energy of the 1b1 MO is almost constant while the 3a1- MO becomes a bonding MO upon bending. The overlap of the 2pz- AOs from A with 1s- AOs from the H- atoms is zero in the linear case, but reducing the angle results in a three- center bonding.

8 PART II ADVANCED AND SPECIAL SUBJECTS 8 The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Figure 3: Walsh diagram for a AH 2 molecule Approximate Intermolecular Interactions In this section we will indicate how the energies and shapes of molecular orbitals can be related to chemical reactivity. One of the key features of chemical reactions is the activation energy that has to be overcome by the reactants. On the basis of perturbation theory and EHT, Klopman and Salem [G. Klopman, JACS, 90, 223 (1968); S. Salem, JACS, 90, 543 (1968)] developed a simple but highly useful decomposition scheme for the interaction energy ΔE of two approaching molecules A and B into contributions from occupied and unoccupied orbitals. All quantities in (10) are obtained for the separate, non- interacting molecules. A,B!E = "#(q µ +q! )" µ! S µ! µ! # + Q I Q J I <J #R IJ # 2( c aµ c q! " µ! ) 2 2( c occ unocc bµ c p! " µ! ) 2 occ unocc µ! µ! + # # + # # a q E a "E q b p E b "E p ( 10) This expression although somewhat lengthy is already simplified (the approach does not take electron- electron interaction into account). So let us discuss the terms step by step. The first term is a double sum over all atomic orbitals (AOs) μ of # 8 Kutzelnigg, W., Einführung in die Theoretische Chemie Band 2, VCH: Weinheim, 1994

9 PART II ADVANCED AND SPECIAL SUBJECTS 9 molecule A and ν of molecule B. The q s are the AO occupation numbers of the non- interacting molecules (in the original paper called charge densities), β is similar but not identical to the resonance integral of Hückel theory (it includes only the electron- nuclear attraction terms), and S is the overlap integral mentioned previously. Since β is always negative, this term is positive and is basically responsible for the occurrence of activation barriers within this model. Its origin is the repulsive interaction between doubly occupied molecular orbitals (MOs) of A and B which arises from the non- symmetric level splitting (Figure 4). Figure 4: Interaction of Two Occupied Molecular Orbitals The second term, a sum over all atoms I of molecule A and atoms J of molecule B, describes the Coulomb interaction between pairs of atoms with net charge Q at distance R (ε is the dielectric constant). The two last terms include sums over all occupied MOs a of A and unoccupied MOs q of B and vice versa. For each pair of orbitals the term consists of sums of corresponding MO coefficients c weighted with β. The dominator is the MO energy difference. This part describes the stabilization due to formation of orbitals combining occupied A MOs with unoccupied B MOs or vice versa in the complex [AB]*. Figure 5: Interaction of an Occupied and an Unoccupied Orbital.

10 PART II ADVANCED AND SPECIAL SUBJECTS Frontier orbital picture It is exactly this last effect which is widely used in further simplified qualitative approaches to determine chemical reactivity. If the two reacting molecules are similar, the smallest energy differences are those between the highest occupied MOs (HOMOs) and the lowest unoccupied MOs (LUMOs). If it is assumed that the other contributions to the last two terms of (1) are always of similar magnitude, the HOMO- LUMO contributions will dominate. If it is further assumed that the first term is roughly independent from the intermolecular orientation, only two main contributions are left for the determination of the feasibility and, sometimes even more important, the regioselectivity of chemical reactions: the Coulomb term and the MO coefficients of the frontier orbitals, HOMO and LUMO. This concept is used e.g. to estimate the relative reactivity of different compounds C towards a given reactant R via calculation of E(HOMO,C)- E(LUMO,R)+E(HOMO,R)- E(LUMO,C). The frontier orbital approach is also applied to ionic reactions in organic chemistry. In electrophilic or nucleophilic reactions the inorganic ions are classified as hard or soft. A hard nucleophilic has a low HOMO energy, a hard electrophilic has a high LUMO energy, and vice versa. In nucleophilic reactions, a hard anion (e.g. OH -, NH2 - ) will attack preferentially the atom with the highest positive charge Q of an organic molecule. In the frontier orbital picture this is due to the large energetic difference between the anion s HOMO and the molecule s LUMO making the last term in the above equation small so that the Coulomb part prevails. On the other hand, the preferred site for a soft anion (CN -, (CH2,Ph)- C- O - ) is the atom with the largest contribution to the molecule s LUMO The Woodward- Hoffman Rules In reaction theory synchronous organic reactions such as pericyclic reactions would in general take place under preservation of the orbital symmetry. Each occupied MO of the reactants will result in an occupied MO of the products with the same symmetry. Experimentally, there are two ways of activating such reactions: 1. Thermal activation

11 PART II ADVANCED AND SPECIAL SUBJECTS Photochemical activation For the theoretical description, the main difference between these two methods lies in the MOs that are involved in these processes. The photochemical activation will result in the excitation of an electron from a bonding (or nonbonding) into an anti- bonding orbital. The resulting state may have a different symmetry than the ground state and will therefore also have a distinct reactivity (Figure 6). Essential difference between photochemical and thermal reactions must therefore be expected and are also observed in practice. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Figure 6: The orbitals that are involved in the different activation cases. 9 A. Electro- cyclic reactions Electro- cyclic reactions may be described as an isomerisation of open- chain polyens to ring isomers which takes place under thermal or photochemical activation. In order to close the ring an additional σ- bond has to created at the expense of a π- bond. In EHT language, the π- MOs on one fragment have to overlap with correct phases with the orbitals of another fragment in order to form a σ- bond. The example of 1,3- Butadiene is shown above in (Figure 6). In fact, it can be determined in advance, when a suitable constructive overlap of the π- orbitals is possible it 9 Woodward, R. B.; Hoffmann R. Angew. Chem. 81. Jahrgang 1969, 21, 797

12 PART II ADVANCED AND SPECIAL SUBJECTS 12 simply depends on the number of π- electrons. As long as both fragments stay in their electronic ground state 4n (n=1,2,...) π- electrons in the molecule lead ti in- phase arrangement of the terminal π- orbitals, whereas 4n+2 (n=1,2,...) π- electrons result in an out- of- phase arrangement. In order to build up a new σ- bond, the π- orbitals in the molecule with 4n π- electrons have to rotate in the same direction (conrotatory) in order to result in a bonding orbital, while the molecules containing 4n+2 π- electrons are obliged to rotate in opposite directions (disrotatory) (compare Figure 7) For photochemical reactions the requirements for constructive overlap are exactly the reverse of the one described above. Therefore, a system with 4n π- electrons has to rotate in opposite directions, whereas a system containing 4n+2 π- electrons needs to rotate in the same direction. Since the two possibilities lead to different stereoisomers of the products that are formed (Figure 7), one can readily predict whether one has to activate a given reaction photochemically or thermally in order to obtain the desired product. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Figure 7: Example for con- and dis- rotatory rotation of the terminal p z orbitals in case of thermal activation

13 PART II ADVANCED AND SPECIAL SUBJECTS 13 Table 2: Summary of allowed and forbidden electrocyclic reactions based on the Woodward- Hoffman rules. m+n-electro-cyclic reactions Thermal activation ground state Photochemical activation excited state forbidden allowed allowed forbidden M+n = 4q q=1, 2,... disrotatory conrotatory disrotatory conrotatory M+n = 4q + 2 q=1, 2,... conrotatory disrotatory conrotatory disrotatory B. Cycloadditions In concerted cycloadditions two new σ- bonds are built up due to overlap of the π- orbitals of both reactants. The location of the σ- bond is arranged either on the same side of the reacting π- system or on the opposite side. The first case is referred to as a suprafacial reaction and the second case as a antarafacial process. The alignment of the responsive π- orbitals is again dependent on the number of π- electrons and on the form of activation as summarized in Figure 8, Figure 9 and Table 3. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Figure 8: Orbital symmetry of a photochemical cyclic addition.

14 PART II ADVANCED AND SPECIAL SUBJECTS 14 The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Figure 9: Orbital symmetry of a thermal cyclic addition. (a) anti- binding (b) binding, but geometrically difficult

15 PART II ADVANCED AND SPECIAL SUBJECTS 15 Table 3: Summary of the Woodward- Hoffman rules. m+n-cyclic addition Thermal activation ground state Photochemical activation excited state forbidden allowed allowed forbidden m+n = 4q q=1, 2,... Π m a + Π n a antara, antara Π m s + Π n a supra, antara Π m s + Π n s supra, supra Π m a + Π n s antara, supra m+n = 4q + 2 q=1, 2,... Π m a + Π n s antara, supra Π m s + Π n s supra, supra Π m s + Π n a supra, antara Π m a + Π n a antara, antara Limitations of the Frontier Molecular Orbital Approach Although the above scheme has been proven to give qualitatively correct predictions in many cases, as we will see below, it is of course limited due to its approximate nature. From the quantum- chemical viewpoint, it is mainly the use of atomic charges and LUMO energy and shape that makes the simple approach questionable. Atomic charges are no observables and strongly depend on the analysis and the basis set. But also the shape of the orbitals is method and basis set dependent. The MOs employed in the above analysis are obtained with minimal basis sets in the extended Hückel framework. Nowadays, one of course wants to go beyond this simple level of theory and use HF or DFT calculations as basis for analysis of chemical reactivity. And here arises a fundamental problem: For a given method, the HOMO converges to something definite while the unoccupied orbitals do not. Remember that only the occupied orbitals, either included in the Slater determinant in HF theory or used to calculate the electron density in DFT, are optimized by the variation procedure. The only criterion for unoccupied orbitals is their orthogonality to the occupied space. In the limit of a complete basis set, the unoccupied space forms a continuum and the LUMO is not really defined. We will demonstrate this in an example in this experiment. In HF theory even the energies Ep, Eq of the unoccupied MOs are fundamentally wrong because the fictitious (virtual) electrons in the UMOs experience the repulsive field of all N electrons of

16 PART II ADVANCED AND SPECIAL SUBJECTS 16 the system, whereas the occupied orbitals experience the correct N- 1 electron potential. Their energy can be related to physical properties (ionization energies) via the Koopmans theorem. The situation in DFT is not much better. Here the virtual MOs experience the correct N- 1 electron repulsive field, but the energies Ea, Eb of the occupied MOs are incorrect due to artificial self- interaction of the electrons which is caused by the approximate description of electron exchange in DFT. As a workaround of the above mentioned shortcomings of HF and DFT, hybrid methods such as B3LYP have been developed where the electron exchange is described by a mixture of the exact one- determinant expression provided by HF theory and a density functional. The mixing coefficient is usually treated as an empirical parameter. Its value is determined by optimal reproduction of experimental properties (including ionization energies and optical spectra). In this way one uses the cancellation of the inherent errors of both approaches. Having this in mind, it is not surprising that the frontier orbital approach fails in some cases. Nevertheless, as a qualitative tool to understand fundamental reasons for the course of chemical reactions, it is still worth to discuss. 1.2 Description of the Experiment Frontier Molecular Orbital Theory As an example we study the reactivity of the pyridinium cation shown below: Figure 10: The pyridinium cation studied in this experiment. Use the Gaussian and/or ORCA programs and one of the methods RHF, B3LYP, or BP86 to calculate the most reactive sites of the pyridinium cation in a reaction with a hard and with a soft nucleophilic, respectively, based on an analysis of the Mulliken net charges and the LUMO coefficients.

17 PART II ADVANCED AND SPECIAL SUBJECTS 17 Create the molecular structure as described in previous experiments, perform a structure optimization and analyse the wavefunction of the final structure. Check if the Mulliken net charges and the shape of the LUMO change if you increase the basis set from SVP to TZVP. Will the qualitative results of the frontier orbital analysis change? Hint: instead of analysing the MO coefficients it is sufficient to draw an orbital picture using MOLDEN or MOLEKEL. In any case the keyword POP=FULL is needed in Gaussian (ORCA prints a Loewdin analysis of each MO by default). Experimentally it is known that hard nucleophilics (OH -, NH2 - ) prefer C(2), and soft nucleophilics (CN -, (CH2,Ph)- C- O - ) prefer C(4). Verify the relative reactivity of the two sites versus OH- and CN- by calculating the energy difference of the four transition structures CN(2), CN(4), OH(2) and OH(4) as shown below. In this case it is sufficient to fully optimize the structures, no TS search is necessary. Figure 11: Structures of the Reaction Products to be Optimized The Woodward- Hoffman Rules - Dissociation of formaldehyde Use the ORCA program to optimize the following structures employing HF/6-311G*

18 PART II ADVANCED AND SPECIAL SUBJECTS 18 and generate.cube files for each MO using orca_plot. Alternatively, you can use the Gaussian program employing HF/6-311G* and generate.chk files. A. H2C=O B. CO C. H2 Classify the HF valence orbitals and the lowest unoccupied orbitals according to their irreducible representation (irrep). Investigate the orbital symmetry concerning the symmetry elements of the point group C2v. Identity (E) two- fold rotation axis (C2) mirror- plane in the xz- plane (σxz) mirror- plane in the yz- plane (σyz) The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Figure 12: Coordinate system used in this experiment. Table 4: The character table of the C 2v point group. C 2v E C 2 σ xz σ yz A z x 2, y 2, z 2 A R z xy B x, R y xz

19 PART II ADVANCED AND SPECIAL SUBJECTS 19 B y, R x Draw a MO diagram for H2C=O on the one side and for CO + H2 on the other side. Connect orbitals which belong to the same irrep with each other. Repeat these steps in Cs symmetry with the symmetry elements E and σ. Remember that there are two possibilities for the orientation of formaldehyde in this point group. What should attract your attention? Walsh s Rules Perform calculations on the following molecules with the Hartree- Fock method and the SVP basis set and the ORCA program: LiH2 + (distance= 1.7 Angström) BeH2 (distance= 1.34 Angström) CH2 (S=0; RKS calculation) (distance=1.078 Angström) CH2 (S=1; ROHF calculation; keyword! ROHF) (distance Angström) H2O (distance=0.956 Angström) First look at the variation of the total energy as a function of angle by performing a rigid scan of that angle and plot the results with xmgrace. An input is: # Scan the angle! RHF SVP %paras Ang= 180,90,17 end * int 0 1 O H H {Ang} 0 * A summary of the total energy as a function of angle will be printed at the bottom of the output file. Now look at the orbitals and their energies at the bent geometry and the linear geometry. Plot them. Discuss the variation of the orbital energies and the total energy as a function of bond angle. Do you find Walsh s results to be consistent with the ab initio results? ADDITIONAL CALCULATIONS (VOLUNTARY)

20 PART II ADVANCED AND SPECIAL SUBJECTS 20 Plot the variation of the nuclear repulsion energy, the one- electron energy, the two- electron energy as well as the kinetic energy of the system as a function of angle. Also plot the sum of the occupied MO energies weighted by their occupancy as a function of bond angle. These data must be extracted from the ORCA output file. What do you find? What is the physical reason for obtaining a bent geometry versus a linear geometry?.