Molecular Orbitals for Ozone

Transcription

1 Molecular Orbitals for Ozone Purpose: In this exercise you will do semi-empirical molecular orbital calculations on ozone with the goal of understanding the molecular orbital print out provided by Spartan and MOPAC at the MNDO level. The term semi-empirical means that some of the integrals necessary in the calculation are estimated from experimental data. The term MNDO pertains to which integrals are approximated and the set of molecules whose experimental data was used to parameterize the integrals. Spartan, Gaussian, and MOPAC do the same calculation; the graphics in Spartan are wonderful, but MOPAC does some analysis of the molecular orbitals that is very useful that Spartan and Gaussian can't do. The print out for the programs is very similar; you will find it easy to switch between the programs. In this Introduction we will use BF 3 as an example. As you read the BF 3 print out see if you can answer the questions: (1) What is the LUMO in BF 3? The LUMO will be the orbital that interacts most strongly with a Lewis base (Lewis bases are electron rich so they need a place to put the extra electrons). (2) What is the charge on the B atom? (3) Is there any π-bonding in the molecule? Introduction The Input Matrix The input matrix for MOPAC calculations can be generated by hand using an editor like Word. For complicated molecules we always use a graphics oriented front end like Spartan; however, using these programs you will never see the input matrix. It is a good idea to know what goes into the input matrix. For example, you don't specify who is bonded to whom; you only specify atom positions. The MOPAC input file for BF 3 is given below. The keywords (appendix 1) on the first line specify that the calculation will be done at the AM1 level and the eigenvectors (wavefunctions), the sigma-pi decomposition, and bond orders will be printed. The "1" after each coordinate indicates that the bond length or angle will be optimized (a "0" specifies no optimization). The first coordinates are distances between atoms, the second and third coordinates are angles. The set of integers on the right specify the relationship of the atoms. The first atom, B, is placed at the origin. The second atom, F, is placed 1.307Å along the x-axis from the first. The third atom, F, is 2.268Å away from atom 2, with an angle of 30 with the y-axis. The fourth atom, also F, is placed 1.307Å away from atom 1, with an angle of 120 from atom 2 and a dihedral of -180 with atoms 3, 1, and 2. The comments at the right, below, are not part of the input file. Note that the atom relationships are purely geometrical; they do not imply the atoms are bonded. AM1 VECTORS PI BONDS BF3 am1 Comments: keywords comment B atom 1 at 0, 0, 0 F atom 2 distance to atom 1 F atom 3 distance to atom 2 and angle with atom 1 F atom 4 has a dihedral of 180 with atoms optimize optimize optimize atom relationships

2 The MOPAC output file, in condensed form, is reproduced below. Some of the tables have also been shortened to save space. The orbitals in the right-most column are not part of the print out. They are included to help you understand and visualize the molecular orbitals. Make sure you understand how the molecular orbital coefficients are determined and their meaning. Draw similar diagrams for other orbitals in the print out. FINAL HEAT OF FORMATION = KCAL TOTAL ENERGY = EV ELECTRONIC ENERGY CORE-CORE REPULSION = = EV EV IONIZATION POTENTIAL = NO. OF FILLED LEVELS = 12 F 3 y x CARTESIAN COORDINATES NO. ATOM X Y Z 1 B F F F EIGENVECTORS ROOT NO S B PX B PY B PZ B S F PX F PY F PZ F F 4 B 1 F 2 atom numbers shown as subscripts S F PX F PY F PZ F Ψ4= σ1 S F PX F PY F PZ F Before continuing with the print out, first note that the molecular orbital coefficient's are listed in columns. For example for orbital 4, which has energy kcal/mol: Ψ4 = s(b 1 ) s (F 2 ) p x (F 2 ) s(f 3 ) p x (F 3 ) p y (F 3 ) s(f 4 ) p x (F 4 ) p y (F 4 ) MO 4 is a σ orbital that contributes strongly to the bonding. The coefficients can answer many questions. For example, the %s character on F atom 3 is given by the ratio of the squared coefficients of the s orbital to the total on the same atom( the other F's will be identical): % s ( on F 3 ) = *100 = 21.2% Continuing with the print out:

3 ROOT NO S B PX B PY B PZ B S F PX F PY F PZ F S F PX F PY F PZ F S F PX F PY F PZ F Ψ7= π1 π1 =0.46pz(B1) +0.51pz(F2) +0.51pz(F3) +0.51pz(F4) ROOT NO S B PX B PY B PZ B S F PX F PY F PZ F S F PX F PY F PZ F S F PX F PY F PZ F NET ATOMIC CHARGES AND DIPOLE CONTRIBUTIONS ATOM NO. TYPE CHARGE ATOM ELECTRON DENSITY 1 B F F F SIGMA-PI BOND-ORDER MATRIX S-SIGMA P-SIGMA P-PI S-SIGMA P-SIGMA P-PI S-SIGMA P-SIGMA P-PI B 1 B 1 B 1 F 2 F 2 F 2 F 3 F 3 F 3 S-SIGMA B P-SIGMA B P-PI B S-SIGMA F P-SIGMA F P-PI F S-SIGMA F P-SIGMA F P-PI F

4 BOND ORDERS AND VALENCIES B 1 F 2 F 3 F 4 B F F F Interpreting Molecular Orbital Print Outs Molecular Orbital Drawings: First note that the orientation of the molecule is with atom 2, which is a F, along the x-axis. The other F's are also in the x-y plane. In picturing the orbitals, start with orbital 7. Since orbital 7 involves p z -orbitals, it is a π type orbital. Note the form of the orbital written in the right hand column. The orbital coefficients are read down the column in the print out. All the p z -orbitals have the same sign, so they all point in the +z direction. Therefore, this is a four-center π-orbital with constructive overlap between all B-F atom pairs. Molecular orbital 4 is a little more difficult to picture. The drawing in the right-hand column has omitted the s orbitals on O for clarity (the overlap of the s-orbital on B with the s-orbitals on the F's is slightly anti-bonding in character, so the s-s overlap doesn't play a big role in the bond formation, anyway). Note that both p x and p y -orbitals on F atoms 3 and 4 participate. Both must be used since the F atom centers are at an angle with the x and y axis. Note that this orbital mixing agrees with hybridization predictions, which predicts that the B will be sp 2 hybridized with the two p- orbitals in the sp 2 mixture being the p x and p y. Here are some of the other things you should be able to glean from the orbital print out: Core Orbitals: You can often spot core orbitals by (1) their unusually low energy and predominate s-character. A second way to spot core orbitals is that (2) they are often nonbonding in character. A third way is to (3) find a set of low energy orbitals that give a total bond order of zero; these orbitals are often in a completely filled bonding-nonbonding-antibonding set. It is arguable if BF 3 has any core orbitals, but considering method 1 above, orbitals 1, 2 and 3 appear to be core orbitals predominately on the F's: note the energy gap between orbital 3 and 4 as compared with the energy gap between orbitals 4 and 5. Considering method 2 above, orbitals 1, 2, and 3 are all bonding in character for BF 3, which argues against them being core orbitals. Considering method 3 above, note that orbitals 1, 2, and 3 are all bonding, which again argues against them being core orbitals. BF 3 is an unusual case, with some ambiguity about the existence of any core orbitals. Many other molecules are easier to deal with, however. You will find ozone to be an easy case with a very clear indication of core orbitals. Sigma and Pi Orbitals: All the orbitals except 7, 8, 9, and 13 involve p orbitals in the plane of the molecule. So 7, 8, 9, and 13 are π type while all the others are σ type. Non-Bonding Orbitals: Of the π orbitals, 7 is bonding and 8 and 9 are non-bonding. You can spot non-bonding orbitals by the absence of orbitals on adjacent atoms. For example, neither 8 or 9 has orbitals on the central B. Don't always expect zero coefficients on adjacent atoms, however. Sometimes non-bonding orbitals have small coefficients on adjacent atoms (eg. ozone), but the shift in the orbital energy is small compared to the atomic energy of the orbital. Anti-bonding Orbitals: Once you have identified a bonding orbital, look for a high energy orbital where the orbital coefficients on an adjacent atom are switched. For example π-orbital 7 has an

5 anti-bonding pair in π*-orbital 13. Note that on orbital 7 the F p z -orbitals have positive coefficients while for orbital 13 the F p z -orbitals have negative coefficients. Another example is the σ* anti-bonding pair to orbital 4. Try to find it before reading further. (The answer is orbital note that the s-orbital on B has the opposite sign but the p-orbitals on the F's have the same sign as in orbital 4.) HOMO and LUMO's: Since BF 3 has 24 electrons, the HOMO will be orbital 12 and the LUMO will be orbital 13. The HOMO is essentially degenerate with orbital 11, and both are nonbonding σ type p-orbitals on the F's. The LUMO is the π*-orbital. The lowest energy electronic absorption transition for this molecule is (LUMO HOMO) a π* nb transition. Atomic Charges: The atomic charges on each atom are printed. As expected the B atom is slightly positive since the F s are more electronegative than B. However, some caution is due: there is no unique way to calculate the charge on an atom. The calculation above tries to predict the charge the atom would have if it were a point charge--which it is not. Different calculation methods will give very different results, so take atomic charges with a grain of salt. Sigma-Pi Bond Order Matrix: The sigma-pi decomposition allows us to see how each atomatom interaction is split into σ andπ-bonds. The resulting "density matrix" is composed of the following basis-functions: s-sigma, p-sigma, p-pi, d-sigma, d-pi, d-dell. The on-diagonal terms give the hybridization state, so for example an sp 2 hybridized system is represented as s-sigma 1.0, p-sigma 2.0, p-pi 1.0. The off diagonal terms give the bond order. For BF 3 the hybridization falls a little short of sp 2 : the hybridization is s p or about sp 1.5. In other words, the in-plane p-orbitals on B don't contribute as much as expected. More interesting is the π bond order of (look in the P-PI B1 column and the P-PI F 3 row of the matrix). Halogens don't usually have large π bond orders. However, F is expected to be unusual since it is at the top of its group and therefore much smaller than the atoms in the later periods. So it isn't surprising that F is different from Cl, Br, and I. (Note that the π bond order in ab initio calculations isn't as large, so the semi-empirical calculation over-estimates π-bond order.) Bond Orders and Valencies: The unusual π bonds add in to make the total bond order between B and F The total valence of the B atom is also increased to 3.32 from the expected 3. In other words the π bonds help to stabilize the molecule. The B-F σ-orbitals withdraw electron density from the B, giving a net positive charge, but the π bonds put back some of the electron density into the empty B p z -orbital. We say that the B atom is a sigma-donor and a pi-acceptor in this kind of bonding. Sigma-donor/pi-acceptor interactions are also important for transition metal complexes. Procedure Outline: Before you come to lab, draw the Lewis dot structure for ozone and also sketch the molecular orbitals for bent and linear ozone using the qualitative rules we have developed in class. Hints: linear ozone will have the same MO's as CO 2, but the MO's will have two more electrons and less s-character. For bent ozone, assume a 90 bond angle and use only p-orbitals on each oxygen.

6 Obtain the print out of the molecular orbitals for ozone from MOPAC. For running MOPAC on a Mac or Windows system see the MOPAC diatomic instructions handout. Do the calculation for bent and linear ozone. For MOPAC it is best for this exercise to build the input matrix by hand, rather than using a graphics front-end to generate the input file. Building the input matrix by hand has some advantages because you can orient the molecule in any way you want. Spartan and similar programs always place the atoms symmetrically about the axes. As you build the input matrix by hand, place the O atoms on the x and y axes and choose the bond angle to be 90 or 180. These coordinates will make the molecular orbital print out much easier to understand and the orbitals easier to draw. For 180 -linear and 90 -bent ozone you must constrain the O-O-O. In MOPAC you can specify not to optimize the angle, while still optimizing the bond lengths; you should put "0" after the angle coordinate rather than a "1", as shown below. MNDO TRIPLET GEO-OK VECTORS BONDS PI Ozone linear O O O This file will be available on the Web for you to edit. It is easiest to use TextEdit on a Mac. For 90 make sure to specify the MNDO GEO-OK VECTORS BONDS and PI keywords. Linear O 3 is a triplet and bent O 3 is a singlet. To constrain the angle in Spartan use the Constrain angle option in the Builder and click the Contraints box in the calculation Setup window. Of course, you can have Spartan generate molecular orbital surfaces to help you visualize the different molecular orbitals. Answer the following questions for bent ozone: 1. Draw the orientation of your ozone relative to the x, y, and z axis for the MOPAC printout. 2. Which are the "core" orbitals? 3. Which are the π orbitals? 4. Which are the non-bonding orbitals? 5. For orbital 4, write out the full molecular orbital, similar to the examples for BF 3 above. 6. Draw orbitals 4, 5, 6 and 10. Follow the BF 3 right-hand column examples above for style.

7 7. In your qualitative MO sketches the non-bonding orbitals will have the same energy. Is this true in fact? 8. What is the hybridization for the central oxygen? 9. What is the π bond order for each bond? 10. Your qualitative model of ozone will start with orbital 4. What is the %s character on the central atom in orbital 4 and orbital 5? Use the eigenvectors (molecular orbital coefficients) for these calculations. Your qualitative model has no s-character in these orbitals. 11. Identify the HOMO and LUMO. What kind of orbitals are they: σ or π type, bonding, nonbonding or anti-bonding orbitals? For linear ozone answer the following questions: 12. Which is predicted to be more stable, linear or bent ozone? 13. Is the π bond order similar in bent and linear ozone? 14. Is the hybridization similar on the central O atom for bent and linear ozone?

8 Access to MOPAC for Larger Molecules The input matrix for MOPAC calculations can always be generated by hand using an editor like Word, SimpleText, or Notepad. For complicated molecules we always use a graphics oriented front end like Spartan, the zmatrix application ( MOE, distance geometry (iris12.colby.edu/~www/jme/dg.html), Chem3D, or using the Macintosh or Windows program PCMODEL. Each access method has its advantages. Spartan, GAMESS, and Gaussian don t actually use the MOPAC program, they implement the MNDO, AM1, and PM3 Hamiltonians within their own native code. The big advantage of MOPAC is access to the sigma-pi decomposition and the NBO routines. Graphics Oriented Front-Ends for MOPAC Method 3D-Structures MO Plots Gaussian Platform Comments zmatrix.html Web PC,Mac 8 atoms dg.html Web PC,Mac Distance geometry MOE PC,Mac QSAR desciptors WebMO Web-Java PC,Mac GAMESS too Chem3D PC,Mac For ChemDraw users PCModel PC,Mac General desktop access There are several other graphical user interfaces to MOPAC available on the Internet.

9 Appendix 1 Some MOPAC Keywords MOPAC calculations are controlled by keywords that appear on the first line of the input file. The following list gives the descriptions of just a few. Please see the MOPAC documentation for a complete listing. The book by Tim Clark also gives an excellent introduction to doing MOPAC calculations: Tim Clark, A Handbook of Computational Chemistry: A Practical Guide to Chemical Structure and Energy Calculations, Wiley, New York, NY, ******Caution: You need to know the spin multiplicity of your molecule. Most ground state molecules are singlets. The default calculation method of Restricted Hartree-Fock theory (RHF) is fine for these molecules. However, if you have an odd-electron molecule or ion, the ground state is almost certainly a doublet. If you are working with excited states or transition states the spin multiplicity can vary. When working with molecules that are not ground state singlets, UHF calculations are the best method to use. However, you can use the DOUBLET, TRIPLET, and EXCITED keywords to stay within the RHF method. These RHF calculations are not as accurate, but they may be sufficient for back-of-the envelope calculations and they are certainly much faster. Please see the MOPAC documentation for more information on UHF calculations. ***************************************************************************** AM1 The AM1 method is to be used. By default MNDO is run. In AM1 the Hamiltonian is calculated slightly differently. BONDS The bond order between all pairs of atoms is printed. In this context a bond is defined as the sum of the squares of the density matrix elements connecting any two atoms. For ethane, ethylene, and acetylene the carbon-carbon orders are roughly 1.00, 2.00, and 3.00, respectively. The diagonal terms are the valencies calculated from the atomic terms only and are defined as the sum of the bonds the atom makes with other atoms. The bonding contributions of all M.O.'s in the system are printed immediately before the bonds matrix. Just as an atomic orbital has a valency, so has a molecular orbital. This leads to the following relations: The sum of the bonding contributions of all occupied M.O.'s is the same as the sum of all valencies which, in turn is equal to two times the sum of all the bonds. The sum of the bonding contributions of all M.O.'s is zero. CHARGE=n When the system being studied is an ion, the charge, n, on the ion must be supplied. For cations n can be 1 or 2 or 3, etc., for anions -1 or -2 or -3 etc. For example for NH 4 + CHARGE=1, and for PO 4 3- CHARGE=-3. DENSITY At the end of the job, when the results are being printed, the density matrix is also printed. If the density is not requested, then the diagonal of the density matrix, i.e., the electron density on the atomic orbitals, will be printed.

10 DOUBLET When a configuration interaction calculation is done, all spin states are calculated simultaneously. When only doublet states are of interest, the DOUBLET can be specified, and all other spin states, while calculated, are ignored in the choice of root to be used. DOUBLET has no meaning in a UHF calculation. EXCITED The state to be calculated is the first excited open-shell singlet state. This state would normally be the state resulting from a one-electron excitation from the HOMO to the LUMO. MOPAC automatically uses a configuration interaction, CI, calculation for excited states. See the MOPAC manual if your molecule is not a ground state singlet. FORCE The vibrational frequencies of the normal modes in a molecule are calculated. These frequencies can then be used to predict the infrared spectrum of the molecule. The force constants are first calculated from the second derivatives of the energy with respect to displacements of all pairs of atoms. The force constants and the isotopic masses are then used to calculate the vibrational frequencies. PI The normal density matrix is composed of atomic orbitals, that is s, px, py, and pz. PI allows the user to see how each atom-atom interaction is split into sigma and pi bonds. The resulting "density matrix" is composed of the following basis-functions: s-sigma, p-sigma, p-pi, d-sigma, d-pi, d-dell. The on-diagonal terms give the hybridization state, so that an sp 2 hybridized system would be represented as s-sigma 1.0, p-sigma 2.0, p-pi 1.0. POLAR The polarizability is calculated. The polarizability allows you to calculate the strength of London dispersion forces between molecules. SINGLET Do a configuration interaction calculation for the ground state singlet of the molecule. TRIPLET The triplet state is defined and a configuration interaction (CI.) calculation is performed. The occupancy of the M.O.'s in the SCF calculation is defined as (...2,1,1,0,...), that is, one electron is put in each of the two highest occupied M.O.'s. ( TRIPLET has an alternate effect in a UHF calculation.) UHF The unrestricted Hartree-Fock Hamiltonian is to be used. UHF calculations are the best method to use for open shell system, i.e. ground state doublets, triplets, etc., and excited states. VECTORS The eigenvectors are to be printed.