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  2. The molecule is a set of atoms that are in strong chemical connection to one another building a new substance. Three types of the strong interactions exist between atoms: • Several individual atoms build the system. Each of them • add electrons to the full system. These electrons are delocalizedandmove practically without resistance in the system: the metal bond was built. 2. One of the interacting atoms has low first ionization energy (e. g. an alkali metal), the second one has high electron affinity (e.g. a halogen element): easy electron transfer from the first to the second atom. Ion pair, they attract each other: we have an ionic bond.

  3. 3. Atoms with open valence shells share a part of their valence electrons, a new electron pair is formed. They build a chemical bond, the covalent bond.Another possibilty: an atom has a non-bonded electron pair, the other an electron pair gap. The electron pair will be common and build a chemical bond, the dative bond. The shared electron pair of the molecule moves on molecular orbitals(MO). Polar bond:the participition of the electron pair between the two atoms is unequal. Delocalized orbital: the bonding electrons are shared under more than two atoms. Molecule: finite number of atoms with the exclusion of polymers. Model: isolated molecule.

  4. Symmetry elements and symmetry conditions Object is symmetric if there exist an operation bringing it in equivalent position. Equivalent: covers the original one. The operations fulfilling this conditions are symmetry operations. Symmetry operations belong to symmetry elements of the object. Symmery elements are mirror planes, symmetry centers (inversion), symmetry axes (girs), reflection-rotation axes (giroids)

  5. Table of symmetry elements and operations

  6. Symmetry elements of water

  7. Demonstration of a tetragiroide

  8. Tetragiroide of methane

  9. Inversion center of trans-hydrogenperoxide

  10. Point groups • Symmetry operations build algebraic groups (G). An algebraic groupis a heap of objects, properties or ideas, characterized as follows. • - A group operation exists. The group is closed for it, the result is member of the group. • a unit element exists (E), X*E=E*X=X • each X element has its inverseY, X*Y=Y*X=E, Y=X-1, X=Y-1 • associativity: (A*B)*C=A*(B*C)A,B,C • conjugate of an element: Y=Z*X*Z-1 and X=Z-1*Y*Z . X,Y,Z A set of the group elements that are conjugated each other builds a class of the group.

  11. Representations of point groups The planar water molecule has two s symmetry (mirror) planes, perpendicular each other. Their crossing axis is a digir, C2. With a unit elementE the build the C2v point group.

  12. There are numbers or matrices those follow this algorithm. They are the representations of the group. The possible simplests of them are the irreducible representations of the group. In spectrocopy they are often called as symmetry species or simply species. If the representation is a matrix, the character table contains the traces of the matrices. The number of representations is equal to those of the classes.

  13. Representations are generally labelled with G. The used notations: Indexing of these symbols:

  14. The rows (i) are species, the columns (j) are classes. For more complicated groups the classes contain more than one element. The table contains the cij coefficients. The gir character is +1, the species is A. If it is -1, the species is B. If sxz character is +1, the subscript is 1, otherwise, if it is -1, the subscript is 2.

  15. Symmetry operations transform atoms in new positions: Proper operations (Cn, E) can be regarded as rotations: Improper operations(Sn, s, i) are rotations + perpendicular reflections:

  16. The traces characterize the transformation matrices, they are independent of the choice of the coordinate system. They are the characters of the symmetry operation: cj for the jth operation. The symmetry of the molecules plays important role in the interpretation of molecular spectra.

  17. The electronic stucture of molecules Construction of molecular orbitals The Born-Oppenheimer theory is used: adiabatic approach. The motion of nuclei are neglected, only the electrons move. The relativistic effects of the Hamilton operator are here neglected: Z is the atomic number, N is the number of atoms, n is the number of electrons in the molecule, r is the distance of the particles. First term: kinetic energy operator, second: electron-electron repulsion, third: electron-nucleus attraction, fourth: nucleus-nucleus repulsion (costant!). The 2nd-4th terms give the potential energy operator. Dis thenabla operator.

  18. Solution of the Scrödinger equation: exactly only for • Additional approximations (restrictions): • Molecular wavefunction: production of molecular orbital functions, depending on the cartesian and spin coordinates; • Pauli’s principle must be satisfied: (Slater) determinant wavefunctions are used, • model of independent particles: each has own orbital (ci) functions,depending only on their own Cartesian coordinates (xi).

  19. Hartree-Fock (HF) method in Roothaan (HFR) representation : orbital functions expanded into series using basicfunctions. Usually atomic orbitals are applyed in praxis. Linear combinations of atomic orbitals as molcular orbitals: LCAO-MO. • Solution of the eigenvalue equations for using the listed approaches: • Self consistent field (SCF) method, it is iterative. Estimating or assuming values for linear coefficents for linear combinations, energy is calculated. With this energy new coeffcients can calculated, with them one have new energy value, etc., until the deviation between the energies of two successive steps arrives the wanted limit. • Shorthand: LCAO-SCF

  20. The symmetry of molecular orbitals Molecular orbitals have symmetry. The orbital functions maybe symmetric: their sign does not change under the under effect of the symmetry operation, cij=+1 (character table); antisymetric, their sign changes under the effect of the symmetry operation, cij=-1 (character table); The symmetry of the molecular orbitals are denoted according to their symmetry species, but lower case letters are used. If some belongs to the same species, they are numbered beginning with them of lowest energy and are used as coefficients.

  21. Orbitals of water molecule. 1 eV = 96.475 kJ mol-1 (LCAO-MO calculations) Filled ring: + region Empty ring: - region Possible bond participations: 1b1 and 2a1

  22. Localized molecular orbitals The LCAO-MO results reflects the electronic structure, but are delocalized. However, they are not suitable for demonstration of the spatial distribution of the electronic structure. The spatial distribution can be introduced with localized orbitals. The linear combination of the localized orbitals have symmetries like the localized ones. They are demonstrative, however, since they are not resuls of quantum chemical calculations, one cannot speak about their energies.

  23. Localized orbitals of water molecule (oxygen orbitals), filled: out-of-plane

  24. Examining the localized orbitals the tetrahedral formation of the four electron pairs around the closed shells of the oxygen atom is well observable. The binding orbitals are less localized than the non-binding orbitals. The 1a1 orbital remains unchanged, i.e. localized. The 1s orbitals of the hydrogen atoms and the 2s, 2px and 2py orbitals of the oxygen atom build the chemical bonds (2a1 and 1b1). There exist also real delocalizedmolecular orbitals, e.g. those of the aromatic rings. Here is difficult to form localized orbitals.

  25. The covalent bond The characteristics of the chemical bond Influences on the formation of the molecular orbitals:(look also at the Hamilton operator) - kinetic energy: smaller free space for electrons higher; - electron-electron repulsion: increases their distance; - electron-nucleus attraction: acts on the electron; - nulceus-nucleus repulsion: important role in the formation of molecular geometry; - spin-spin electron interaction: with parallel spin repulsive, with opposite spin attractive(Pauli principle, Hund rule).

  26. Results for localized elecron pairs: • -Try to avoid one another; • Try to expanding their possible area, • Try to come as close to the nucleus as possible The molecular geometry is the result of the listed effects. Formation of the molecular orbital: the electron clouds of the atoms approach one another. Hybridization: mixing of the atomic orbitals (y):overlap integral, measure (grade) of mixing (atoms A and B):

  27. Mixing of atomic orbitals: chemical bond. • Extreme cases: • there is not mixing of atomic orbitals, e.g. water 1a1 orbital; • the participations are equivalent, like H-H bond in hydrogen molecule, with 1s orbitals. • During the approaching of the atomic orbitals two levels build, these molecular orbitals: • the energy of one is lower than those of the atomic orbials, it is localized between the two atoms, this is the bonding molecular orbital; • the energy of the other is increased, it has a nodal surface, is wide spreaded, this is the antibonding molecular orbital.

  28. The intoduced model is valid only in case of bonds with s-s atomic orbitals. The more atomic orbitals with nearly the same energy levels are combined in the bond, the greater the deformation of the original atomic orbitals. The description of the molecular orbitals is possible only as the linear combination of several atomic orbitals. If only elements with atomic number lower than 10 take part in the molecule, the deformation of the atomic orbitals is small. The attractive force between the interatomic electron clouds and the atomic cores is greater than the repulsive force between the atomic cores (nucleus and inner electrons). This is the fundamental reason of the formation of chemical bonds.

  29. The intramolecular electron affinity of the atoms is characterized by the electronegativity. Under several definitions the widely used if that of Mullikan: Here I is the ionization energy, A is the electronaffinity of the atom. The atoms at the first part of the periodic table having high electronegativity like carbon, nitrogen and oxygen and can mobilize even two or three electrons to fill their valence electron shell. The second and third bonds are weaker than the first one since the interatomic area is occupied by the electron pair of the first bond (repulsion). A multiple bond needs atomic orbitals of appropriate orientation (p or d orbitals) that energy level is not very high.

  30. The structure of two-atomic molecules The simplest molecules, suitable for studying the chemical bond. Two equivalent atoms: point group: , cylindric form Infinite gir, 2 operations, Infinite vertical planes, Inversion center, Infinite giroid, 2 operations, Infinite vertical digirs. Special labels for species of diatomic molecules.

  31. Hetero diatomic molecules have lower symmetry, the symmetry elements of this group are only the gir and the infinite number of mirror planes, cutting the gir. Special labels are applied for symmetry species of diatomic molecules: the Greek letters instead of the corresponding Latin ones. The sigma (s) bond is cylindric, between the two atoms, maybe s-s. s-p or p-p bond. This is the strongest bond. The pi (p) bond is situated out of the interatomic area, maybe p-p, p-d or d-d bond, weaker than the sigma ones.

  32. Look again on the forms of the atomic orbitals! Where are possible s or p bonds?

  33. Molecular orbitals for H2 from 1sA and 1sB atomic orbitals. The antibonding orbitals are starred (*).

  34. Hybridization A molecular orbital is called hybrid orbital if an atom takes part in it with more then one orbitals. Measure: participation of the atomic orbitals in the molecular wavefunction. Hybridization is possible only in case of s bonds. The central atom contacts n equivalent atoms or atom groups. Results: n equivalent orbitals arranging symmetrically in space and determine the structure. Example. The ground state of the C atom is 1s22s2p2().If one 2s electron transits to a 2p orbital (according to Hund's rule) then the electron configuration changes to 1s22sp3 ().In space symmetric, equienergetic orbitals, sp3 hybrids.

  35. These hybrid orbitals are orthogonal to one another (in algebraic sense), therefore their overalap integrals are zero. The four 5S2 hybrid molecular wavefunctions are (atomic orbitals are denoted as c): • Therefore methan has tetrahedral structure. From onecarbon to methan with hybridization (2 ways): • Excitation of C (promotional energy needed), combined with four hydrogens (energy recovered); • C is combined with four H’s, CH4 is in excited state, energy loss to lower 5S2 state.

  36. In the MO theory the hybridization means the forming of equivalent orbitals. Beside this sp3 hybrid orbitals they are formed with ethene (C2H4): sp2 hybrid and also with ethine (C2H2): sp hybrid. The substitution demages these hybrid orbitals since their equivalence disappears. The hybridization is important in case of complex compounds of transition elements. Their d orbitals can form hybrid molecular orbitals with the ligands. E.g.: spd2 determines a square structure [(PtCl4)2-] , sp3d a trigonal bipyramide (PCl5), sp3d2 an octaheder (SF6) , etc.

  37. Delocalized systems Organic compounds with conjugated double bonds are special case of the double bonded molecules. Beside the first s bonds each second chemical bond is strengthened though a p bond. However the electrons of the p bonds spread along the the whole so-called conjugated system The energy levels are far over the s levels. The sp separation is agood approach for describing the system. Restrictions for the simple Hückel method (p levels): 1. for overlap integrals or 2. Hamilton matrix element is denoted as a, i and j on same atom (Coulomb integral); b, i and j on vicinal atoms (resonance integral); is zero, otherwise.

  38. Both constants have negative sign, a is of higher absolute value. The eigenvalue equation has the form For the ethylene (ethene) molecule (only carbon atoms are considered): for bonding p orbital The results: for antibonding p orbital Extended Hückel theory (EHT) for heterocyclic systems: ax=a+hxb, bxy=kxy*b, e.g.hN=0.5, kCN=1.

  39. Application of the Hückel theory to benzene: the results of the eigenvalue equation are (b is assumed as -75 kJ/mol): Two levels are degenerated. The six b electrons occupy the lowest E4, E5 and E6 levels. For one carbon atom E=a+b.Without conjugation is the total energy 6*(a+b)=6a+6b. With conjugationE=2*(a+2b)+4*(a+b) i.e. E=6a+8b. The energy decreased since 2b= -150 kJ/mol, this is the delocalization energy. The Hückel theory is an acceptable approach for such cases.

  40. For the point of view of reactivity of molecules two energy levels are important: The electron density on the highest occupied molecular orbital (HOMO) is nearly proportional to the reactivity in electrophylic reactions. The electron density on the lowest unoccupied molecular orbital (LUMO) is nearly proportional to the reactivity in nucleophylic reactions. These limit levels play also important role in the development of the chemical and spectroscopic properties of the molecule. The advanced quantum chemical methodsresult better approximations, like post-Hartree-Fock and density functional methods (DFT: density functional theory). The d orbitals complicate these calculations.

  41. Complex compounds of the transition metals The d orbitals are important since several transition metals play role in catalysts and enzymes. Their description is more complicate than that of the molecules with atoms below atomic number 10. Even a simple theory is a good tool in this field. Bethe's crystal field theory is simple, old, but suitable also in our days. The ligands with their negative charges (ion or dipole) connect the central ion (having positive charge). The bond is relatively weak. Practically the central ion determines the molecular structure. The electric field acts on the crystal field, the spin-orbital interaction and the internal magnetic field take also part in the Hamilton operator of the molecule.

  42. The discussion of the nd (n>3) and nf orbitals is complicate. Our model is the 3d orbital. Since n=3, the maximal angular quantum number l=2, magnetic quantum number changes from m=-2 to m=2. Octahedral complexes with six equvalent ligands (sp3d2 hybrids) are discussed here. They belong to the Oh point group. The angle depending parts of the d orbital functions determine the symmetry.The and the orbitals (transforming like 3) are symmetric to the xy, xz and yz mirror planes (3sh) and are also symmetrical to the x, y and z digirs (3C2), therefore they belong to the symmetry species Eg.The three other d orbitals, dxy, dxz and dyz are symmetric to the six axis-axis bisectors (6C2) and to the mirror planes determined by a bisector and an axis (6sd) and to the inversion (i). Therefore they belong to T2g.(as labels T and F are equivalent).

  43. Oh karaktertáblázat

  44. The originally five times degenerated energy level splitsinto two groups. The ligandsconnecting to the central ion are positioned on the coordinate axes. The t2g orbitals (orbitals are labelled similarly to their symmetry species only small letters are used) are situated between the coordinate axes, while the eg orbitals are centred on them. Therefore the ligandsrepulse the eg orbitals, so their energy is higher than thatof the t2g ones. The energy difference between these two orbital groups depends above all on the electric field generated by the ligands. The experimentally measured splitting is denoted by D. The crystal field theory gives the order of the orbital energies but only as expressions, their values are not calculable.

  45. The measure of the splitting in octahedral crystal field is labelled by 10Dq. Using the experimental data the Dq becomes calculable (q is the ratio of two matrix elements, D is a coefficient in the description of the crystal field). The shift of the band system by the ligands is not takeninto account. Therefore the average energy of the d orbitals is always 0 D. The splitting is influenced by two effects: 1. The crystal field (metal ion - ligand, d orbital symmetry) effect. 2. The mutual repulsion of the d electrons. First effect stronger:strong crystal field, second effect stronger:weak crystal field.

  46. In a strong crystal field the electrons occupy the energy levels according to the increasing energy. Therefore the t2g orbitals are occupied at first, and the eg ones only later. The energy of the t2g orbitals is 4Dq lower than average, while that of the eg orbitals is 6Dq higher than average. The t2g orbitals are the bonding ones, the eg's are the antibonding ones.

  47. d electron configurations in strong octahedral crystal field

  48. The situation is more complicate in weak crystal fields. The repulsion of the electrons split the orbitals into several levels. Sometimes these levels are very close. The electrons occupy the orbitals according Hund's rule. At first all orbitals are occupied by one electron. After the occupation of all orbitals in this way the second electrons join stepwise the first ones with opposite spins. Octahedral complexes with weak and strong crystal fields differ in the case of d4, d5, d6 and d7 configurations. The group spin quantum numbers of these configurations are for weak crystal fields high, they are high spin states. Forstrong crystal fields the group spin quantum number is in these cases low, they are low spin states. These two types of states are distinguishable by magnetic measurements.