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William A. Goddard, III, wag@kaist.ac.kr

Lecture 9, September 30, 2009. Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy. Course number: KAIST EEWS 80.502 Room E11-101 Hours: 0900-1030 Tuesday and Thursday.

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William A. Goddard, III, wag@kaist.ac.kr

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  1. Lecture 9, September 30, 2009 Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy Course number: KAIST EEWS 80.502 Room E11-101 Hours: 0900-1030 Tuesday and Thursday William A. Goddard, III, wag@kaist.ac.kr WCU Professor at EEWS-KAIST and Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology Senior Assistant: Dr. Hyungjun Kim: linus16@kaist.ac.kr Manager of Center for Materials Simulation and Design (CMSD) Teaching Assistant: Ms. Ga In Lee: leeandgain@kaist.ac.kr Special assistant: Tod Pascal:tpascal@wag.caltech.edu EEWS-90.502-Goddard-L09

  2. Schedule changes, reminder There was no lecture on Sept. 22 because of the EEWS conference Goddard will be traveling Oct 2-11 and will not give the lectures scheduled for Oct. 6 and 8 Consequently an extra lecture will be added at 2pm on Wednesday Sept. 30 and another at 2pm Wednesday Oct 14. This will be in the same room, 101 E11 L8: Sept. 29, as scheduled, 9am L9: Sept. 30, new lecture 2pm replaces Oct 6 L10: Oct. 1, as scheduled, 9am L11: Oct. 13, as scheduled, 9am L12: Oct. 14, new lecture 2pm, replaces Oct 8 L13: Oct. 15, as scheduled, 9am EEWS-90.502-Goddard-L09

  3. Volunteer extra credit assignment For those that will be bored with no lectures, we have a course project for extra credit. The numbers in these notes were mostly put together in 1972 to 1986, when the QM calcualtions were less accurate and the experiments less complete. I would like to update all the numbers using modern QM methods (B3LYP DFT with 6-311G** basis set) of moderate accuracy. Dr. Hyungjun Kim has a number of systems we would like to update at this level and will work with interested students Satisfactory completion of such a project, with a report summarizing the results can be used in lieu of the class midterm. EEWS-90.502-Goddard-L09

  4. Last time EEWS-90.502-Goddard-L09

  5. The role of symmetry in QM In this course we are concerned with the solutions of the Schrodinger equation, HΨ=EΨ, but we do actually want to solve this equation. Instead we want to extract the maximum information about the solutions without solving it. Symmetry provides a powerful tool for doing this. Some transformation R1is called a symmetry transformation if it has the property that R1(HΨ)=H(R1Ψ) The set of all possible symmetries transformations of H are collected into what is called a Group. EEWS-90.502-Goddard-L09

  6. The definition of a Group 1). Closure: If R1,R2 e G (both are symmetry transformations) then R2 R1is also a symmetry transformation, R2 R1e G 2. Identity. The do-nothing operator or identity, R1 = ee G isclearly is a symmetry transformation 3. Associativity. If (R1R2)R3 =R1(R2R3). 4. Inverse. If R1e G then the inverse, (R1)-1e G,where the inverse is defined as (R1)-1R1 = e. EEWS-90.502-Goddard-L09

  7. The degenerate eigenfunctions of H form a representation If HΨ=EΨ then H(R1Ψ)= E(R1Ψ) for all symmetry transformations of H. Thus the transformations amount the n denegerate functions, {S=(RiΨ), where RiΨieG} lead to a set of matrices that multiply in the same way at the group operators. The Mathematicians say that these functions form a basis for a representation of G. Of course the functions in S may not all be different, so that this representation can be reduced. The mathematicians went on to show that one could derive a set of irreducible representations that give all possible symmetries for the H. reorientations from which one can construct any possible. EEWS-90.502-Goddard-L09

  8. Example, an atom. For an atom any rotations about any axis passing through the nucleus is a symmetry transformation. This leads to the group denoted as SO(3) by the mathematicians [O(3) indicates 3 three-dimensional real space, S because the inversion is not included). The irreducible representations of O(3) are labeled as S (non degenerate) and referred to as L=0 P (3 fold degenerate) and referred to as L=1 D (5 fold degenerate) and referred to as L=2 F (7 fold degenerate) and referred to as L=3 G (9 fold degenerate) and referred to as L=4 EEWS-90.502-Goddard-L09

  9. H2O, n example of C2v consider the nonlinear H2A molecule, with equal bond lengths, e.g. H2O, CH2, NH2 • The symmetry transformations are • e for einheit (unity) xx, yy, zz • C2z, rotation about the z axis by 2p/2=180º, x-x, y-y, zz • sxz, reflection in the xz plane, xx, y-y, zz • syz, reflection in the yz plane, x-x, yy, zz • Which is denoted as the C2v group. EEWS-90.502-Goddard-L09

  10. Stereographic projections e e y y x x C2z C2z syz sxz sxz syz syz C2z C2z syz sxz sxz Consider the stereographic projection of the points on the surface of a sphere onto a plane, where positive x are circles and negative x are squares. Start with a general point, denoted as e and follow where it goes on various symmetry operations. This make relations between the symmetry elements transparent. e.g. C2zsxz= syz Combine these as below to show the relationships C2v EEWS-90.502-Goddard-L09

  11. The character table for C2v The basic symmetries (usually called irreducible representations) for C2v are given in a table, called the character table My choice of coordinate system follows (Mulliken in JCP 1955). This choice removes confusion about B1 vs B2 symmetry (x is the axis for which sxz moves the maximum number of atoms In the previous slide we saw that C2zsxz= syz which means that the symmetries for syz are already implied by C2zsxz. Thus we consider C2z andsxz as the generators of the group. This group is denoted as C2v, which denotes that the generators are C2z and a vertical mirror plane (containing the C2 axis) EEWS-90.502-Goddard-L09

  12. Symmetries for H2O, NH2, and CH2 CH2 NH2 H2O A{(O2py)2[(Opx )(Hx)+(Hx)(Opx)](ab-ba)[(Opz)(Hz)+(Hz)(Opz)](ab-ba)} H2O OHx bond OHz bond 1A1 2B1 1A1 NH2 A{(N2pya)[(Npx )(Hx)+(Hx)(Npx)](ab-ba)[(Npz)(Hz)+(Hz)(Npz)](ab-ba)} NHx bond NHz bond CH2 A{(C2py)0[(Cpx )(Hx)+(Hx)(Cpx)](ab-ba)[(Cpz)(Hz)+(Hz)(Cpz)](ab-ba)} EEWS-90.502-Goddard-L09

  13. Now do triplet state of CH2 A{(C2sa)1(2pxa)1[(CpL )(HL)+(HL)(CpL)](ab-ba)[(CpR)(HR)+(HR)(CpR)](ab-ba)} CHL bond CHR bond Since we know that the two CH bonds are invariant under all symmetry operations, from now on we will write the wavefunction as y z A{[(CHL)2(CHR)2](Csa)1(Cpa)1} Here s is invariant (a1) while p transforms as b1. Since both s and p are unpaired the ground state is triplet or S=1 p=2px s=2s Thus the symmetry of triplet CH2 is 3B1 EEWS-90.502-Goddard-L09

  14. Second example, C3v, with NH3 as the prototype z x A{[(Npy )(Hy)+(Hy)(Npy)](ab-ba)[(Npx )(Hx)+(Hx)(Npx)](ab-ba)[(Npz)(Hz)+(Hz)(Npz)] (ab-ba)} NHx bond NHc bond NHb bond We will consider a system such as NH3, with three equal bond lengths. Here we will take the z axis as the symmetry axis and will have one H in the xz plane. The other two NH bonds will be denoted as b and c. EEWS-90.502-Goddard-L09

  15. C3v,prototype: NH3 z x The symmetry transformations of C3v are: Hx • e for einheit (unity) xx, yy, zz • Cz3, rotation about the z axis by 2p/3=120º • (Cz3)2 rotation about the z axis by 2(2p/3)=240º • sxz, reflection in the xz plane, xx, y-y, zz • syz, reflection in the plane through Hb and the z axis • syz, reflection in the plane through Hc and the z axis Hb Hc EEWS-90.502-Goddard-L09

  16. Combining the 6 operations for C3v leads to b sxzC32 C3 e x sxz sxzC3 C32= C3-1 c Clearly this set of symmetries is closed, which you can see by the symmetry of the diagram. Also we see that the generators are sxz andC3 Note that the operations do not all commute. Thus sxzC3 ≠ sxzC3 Instead we get sxzC3 = sxzC32 Such groups are called nonabelian and lead to irreducible representations with degree (size) > 1 EEWS-90.502-Goddard-L09

  17. The character table for C3v Here the E irreducible representation is of degree 2. This means that if φpx is an eigenfunction of the Hamiltonian, the so is φpy and they are degenerate. This set of degenerate functions would be denoted as {ex,ey} and said to belong to the E irreducible representation. The characters in this table are used to analyze the symmetries, but we will not make use of this until much later in the course. Thus an atom in a P state, say C(3P) at a site with C3v symmetry, would generally split into 2 levels, {3Px and 3Py} of 3E symmetry and 3Pz of 3A1 symmetry. EEWS-90.502-Goddard-L09

  18. Application for C3v, NH3 z x sLP We will write the wavefunction for NH3 as x A{(sLP)2[(NHb bond)2(NHc bond)2(NHx bond)2]} c b Where each of the 3 VB bond pairs is written as a pair function and we denote what started as the 2s pair as slp The rotations and relections interchange pairs of electrons leaving the wavefunction invariant. Thus we get 1A1 symmetry EEWS-90.502-Goddard-L09

  19. Linear molecules, C∞v symmetry For a linear molecule (axis along z) the symmetry operators are: e: einheit (unity) Rz(a): counterclockwise rotation by an angle a about the z axis sxz: reflection in the xz plane s’ = Rz(a) sxz Rz(-a); reflection in a plane rotated by an angle a from the xz plane (there are an infinite number of these This group is denoted as C∞v EEWS-90.502-Goddard-L09

  20. The character table for C∞v EEWS-90.502-Goddard-L09

  21. The symmetry functions for C∞v Lower case letters are used to denote one-electron orbitals EEWS-90.502-Goddard-L09

  22. Application to FH The ground state wavefunction of HF is A{(F2px)2(F2py)2[(Fpz)(H)+(H)(Fpz)](ab-ba)} In C∞v symmetry, the bond pair is s (m=0),while the px and py form a set of p orbitals (m=+1 and m=-1). Consider the case of up spin for both px and py Ψ(1,2) = A{φxaφya}=(φxφy- φyφx) aa Rotating by an angle g about the z axis leads to (φaφb- φbφa) = [(cosg)2 +(sing)2] }=(φxφy- φyφx) Thus (φxφy- φyφx) transforms as S. EEWS-90.502-Goddard-L09

  23. Continuing with FH Thus the (px)2(py)2 part of the HF wavefunction A{(F2px)2(F2py)2[(Fpz)(H)+(H)(Fpz)](ab-ba)} transforms as S. The symmetry table, demands that we also consider the symmetry with respect to reflection in the xz plane. Here px is unchanged while py changes sign. Since there are two electrons in py the wavefunction is invariant. Thus the ground state of FH has 1S+ symmetry EEWS-90.502-Goddard-L09

  24. x z Next consider the ground state of OH We write the wavefunctions for OH as 2Px Ψx=A{(sOHbond)2[(pxa)(pya)(pxb)]} Ψy=A{(sOHbond)2[(pxa)(pya)(pyb)]} We saw above that A{(pxa)(pya)} transforms like S. thus we need examine only the transformations of the downspin orbital. But this transforms like p. 2Py Thus the total wavefunction is 2P. Another way of describing this is to note that A{(px)2(py)2} transforms like S and hence one hole in a (p)4shell, (p)3 transforms the same way as a single electron, (p)1 EEWS-90.502-Goddard-L09

  25. x z Now consider the ground state of NH A{(NH bond)2(N2pxa)(N2pya)} We saw earlier that up-spin in both x and y leads to S symmetry. With just one electron in py, we now get S-. Thus the ground state of NH is 3S-. EEWS-90.502-Goddard-L09

  26. Now consider Bonding H atom to all 3 states of C x z Bring H1s along z axis to C and consider all 3 spatial states. (2px)(2pz) O 2pz singly occupied. H1s can get bonding Get S= ½ state, Two degenerate states, denote as 2P (2py)(2pz) (2px)(2py) No singly occupied orbital for H to bond with EEWS-90.502-Goddard-L09

  27. x z Ground state of CH (2P) The full wavefunction for the bonding state 2Px A{(2s)2(OHs bond)2(O2pxa)1} 2Py A{(2s)2(OHs bond)2(O2pya)1} EEWS-90.502-Goddard-L09

  28. x z Bond a 2nd H atom to the ground state of CH Starting with the ground state of CH, we bring a 2nd H along the x axis. Get a second covalent bond This leads to a 1A1 state. No unpaired orbtial for a second covalent bond. EEWS-90.502-Goddard-L09

  29. x z θe Re Analyze Bond in the ground state of CH2 Ground state has 1A1 symmetry. For optimum bonding, the pz orbital should point at the Hz while the px orbital should point at the Hx. Thus the bond angle should be 90º. As NH2 (103.2º) and OH2 (104.5º), we expect CH2 to have bond angle of ~ 102º EEWS-90.502-Goddard-L09

  30. But, the Bending potential surface for CH2 1B1 1Dg 1A1 3B1 3Sg- 9.3 kcal/mol The ground state of CH2 is the 3B1 state not 1A1. Thus something is terribly wrong in our analysis of CH2 EEWS-90.502-Goddard-L09

  31. Re-examine the ground state of Be 2 a0 1s 2s 1.06A 0.14A R~0.14A Ground state of Be atom: A[(1s)2(2s)2] = A[(1sa)(1sb)(2sa)(2sb)] Each electron has its maximum amplitude in a spherical torus centered at R2s ~ 1.06 A = 2.01 bohr Thus the two electrons will on the average be separated by 2*sqrt(2) = 2.8 bohr leading to an ee repulsion of ~1/2.8 hartree= 9.5 eV EEWS-90.502-Goddard-L09

  32. hybridization of the atom 2s orbitals of Be. So assumed that the Be wavefunction is A[(1s)2(2s)2] = A[(1sa)(1sb)(2sa)(2sb)] 2s In fact this is wrong. Writing the wavefunction as A[(1sa)(1sb)(φaa)(φbb)] and solving self-consistently (unrestricted Hartree Fock or UHF calculation) for φa and φbleads to φa = φ2s + lφpz and φb = φ2s - lφpz where φpz is like the 2pz orbital of Be+, but with a size like that of 2s rather (smaller than a normal 2p orbital) This pooching or hybridization of the 2s orbitals in opposite directions leads to a much increased average ee distance, dramatically reducing the ee repulsion. pz φ2s + lφpz φ2s - lφpz EEWS-90.502-Goddard-L09

  33. analyze the pooched or hybridized orbitals x x z z Pooching of the 2s orbitals in opposite directions leads to a dramatic increase in ee distance, reducing ee repulsion. φ2s + lφpz φ2s - lφpz z z 1-D 2-D Schematic. The line shows symmetric pairing. Notation: sz and sz bar or ℓ and ℓ bar. Cannot type bars. use zs to show the bar case EEWS-90.502-Goddard-L09

  34. Problem with UHF wavefunction A[(φaa)(φbb)] = [φa(1)a(1)][(φb(2)b(2)] – [φb(1)b(1)] [φa(2)a(2)] Does not have proper spin or space permutation symmetry. Combine to form proper singlet and triplet states. 1Ψ(1,2) = [φa(1)φb(2)+φb(1)φa(2)][a(1)b(2)–b(1)a(2)] 3Ψ(1,2) = [φa(1)φb(2)-φb(1)φa(2)][a(1)b(2)+b(1)a(2)] and aa, bb The Generalized Valence Bond (GVB) method was developed to optimize wavefunctions of this form. The result is qualitatively the same as UHF, but now the wavefunction is a proper singlet. I do not have handy a plot of these GVB orbitals for Be but there are similar to the analogous orbtials for Si, which are shown next EEWS-90.502-Goddard-L09

  35. The GVB orbitals for the (3s)2 pair of Si atom Long dashes indicate zero amplitude, solid lines indicate positive amplitude while short dashes indicate negative amplitude. The spacing between contours is 0.05 in atomic units EEWS-90.502-Goddard-L09

  36. Analysis of the GVB singlet wavefunction 1Ψ(1,2) = [φa(1)φb(2)+φb(1)φa(2)][a(1)b(2)–b(1)a(2)] Substituting φa = φ2s + lφpz and φb = φ2s - lφpz into the spatial factor leads to (ab+ba) = (s+lz)(s-lz)+(s-lz)(s+lz) = [s(1)s(2) - l2 z(1)z(2)] (ignoring normalization), which we will refer to as the CI form (for configuration interaction). In the GVB wavefunction it is clear from the shape of the sz and zs wavefunctions that the average distance between the electrons is dramatically increased. This is a little more complicated to see in the CI form. Consider two electrons a distance R from the nucleus. Then the probability for the two electrons to be on the same side is s(R)s(R)-l2 z(R)z(R) which is smaller than s(R)s(R) while the probability of being on opposite sides is s(R)s(-R)- l2 z(R)z(-R) = s(R)s(R)+l2 z(R)z(R) which is increased. EEWS-90.502-Goddard-L09

  37. Analysis of the GVB singlet wavefunction 1Ψ(1,2) = [φa(1)φb(2)+φb(1)φa(2)][a(1)b(2)-b(1)a(2)] where φa = φ2s + lφpz and φb = φ2s - lφpz The optimum value of l~0.4 (it is 0.376 for Si) which leads to a significant increase in the average ee distance, but from the CI expansion [s(1)s(2) - l2 z(1)z(2)]/sqrt(1+l4) We see that the wave function is still 86% (2s)2 character. This is expected since promotion of 2s to 2p costs a significant amount in the one electron energy. This promotion energy limits the size of l. Normalizing the GVB orbitals leads to φa = (φ2s + lφpz)/sqrt(1+l2) Thus the overlap of the GVB pair is < φa | φa > = (1-l2)/(1+l2)=0.752, similar to a bond pair EEWS-90.502-Goddard-L09

  38. Analysis of the GVB triplet wavefunction 3Ψ(1,2) = [φa(1)φb(2)-φb(1)φa(2)][a(1)b(2)+b(1)a(2)] and aa, bb Substituting φa = φ2s + lφpz and φb = φ2s - lφpz into the spatial factor leads to (ab-ba) = (s+lz)(s-lz)-(s-lz)(s+lz) = [s(1)z(2)-z(1)s(2)] (ignoring normalization). This is just the wavefunction for the triplet state formed by exciting the 2s electron to 2pz, which is very high (xx eV). Thus we are interested only in the singlet pairing of the two lobe or hybridized orbitals. This is indicated by the line pairing the two lobe functions EEWS-90.502-Goddard-L09

  39. Problem with the GVB wavefunction A problem with this simple GVB wavefunction is that it does not have the spherical symmetry of the 1S ground state of Be. This problem is easily fixed in the CI form by generalizing to {s(1)s(2) - m2 [z(1)z(2)+x(1)x(2)+y(1)y(2)]} which does have 1S symmetry. Here the value of m2 ~ l2/3 This form of CI wavefunction can be solved for self-consistently, referred to as MC-SCF for multiconfiguration self-consistent field. But the simple GVB description is obscured. EEWS-90.502-Goddard-L09

  40. Role of pooched or hybridized atomic lobe orbitals in bonding of BeH+ In fact optimizing the wavefunction for BeH+ leads to pooching of the 2s toward the H1s with much improved overlap and contragradience. Consider the bonding of H to Be+ The simple VB combination of H1s with the 2s orbital of Be+ leads to a very small overlap and contragradience EEWS-90.502-Goddard-L09

  41. Role of pooched or hybridized atomic lobe orbitals in bonding of BeH neutral At small R the H can overlap significantly more with sz than with zH, so that we can form a bond pair just like in BeH+. This leads to the wavefunction At large R the orbitals of Be are already hybridized A{[(sz)(H)+(H)(sz)](ab-ba)(zsa)} zs sz H In which the zs hybrid must now get orthogonal to the sz and H bond pair. This weakens the bond from that of BeH+ by ~ 1 eV Thus the wave function is A{[(sz)(zs)+(zs)(sz)](ab-ba)(Ha)} where sz≡(s+lz) and zs ≡(s-lz) Here the H overlaps slightly more with sz than with zs, but the spin on sz is half the time a Thus at large R we obtain a slightly repulsive interaction. zs sz H EEWS-90.502-Goddard-L09

  42. Short range Attractive interaction sz with H Compare bonding in BeH+ and BeH Long range Repulsive interaction with H BeH TA’s check numbers, all from memory 2 eV BeH+ has long range attraction no short range repulsion 3 eV 1 eV BeH+ 1eV Repulsive orthogonalization of zs with sz H EEWS-90.502-Goddard-L09

  43. Compare bonding in BeH and BeH2 BeH+ MgH+ 1S+ 3.1 eV R=1.31A 2.1 eV R=1.65 A 1.34 eV R=1.73A 2.03 eV R=1.34A 2S+ ~3.1 eV 1S+ linear ~2.1 e Expect linear bond in H-Be-H and much stronger than the 1st bond Expect bond energy similar to BeH+, maybe stronger, because the zs orbital is already decoupled from the sz. TA’s check numbers, all from memory Cannot bind 3rd H because no singly occupied orbitals left. EEWS-90.502-Goddard-L09

  44. New material EEWS-90.502-Goddard-L09

  45. The ground state for C atom Based on our study of Be, we expect that the ground state of C is Modified from the simple (1s)2(2s)2(2p)2 form Ψyz=A[(2sa)(2sb)(ya)(za)] Consider first pooching the 2s orbitals in the z direction Ψyz=A[(2s+lz)(2s-lz)+(2s-lz)(2s+lz)](ab-ba)(ya)(za)] 2s pair pooched +z and –z yz open shell Expanding in the CI form leads to Ψyz=A[(2s)(2s)-l2(z)(z)](ab-ba)(ya)(za)] =A[(2sa)(2sb)](ya)(za)] -l2A[(za)(zb)(ya)(za)] But the 2nd term is zero since the za is already occupied Thus the 2s can only pooch in the x direction sx py pz Ψyz=A[(sx)(xs)+(xs)(sx)](ab-ba)(ya)(za)] xs EEWS-90.502-Goddard-L09

  46. The GVB orbitals of Silicon atom Long dashes indicate zero amplitude, solid lines indicate positive amplitude while short dashes indicate negative amplitude. The spacing between contours is 0.05 in atomic units EEWS-90.502-Goddard-L09

  47. The ground state for C atom x z Based on our study of Be, we expect that the ground state of C is Ψyz=A[(sx)(xs)+(xs)(sx)](ab-ba)(ya)(za)] which we visualize as sx py 2s pair pooched +x and –x yz open shell pz xs Ψyx=A[(sz)(zs)+(zs)(sz)](ab-ba)(ya)(xa)] which we visualize as px py 2s pair pooched +z and –z xy open shell zs sz Ψxz=A[(sy)(ys)+(ys)(sy)](ab-ba)(xa)(za)] which we visualize as px xz open shell 2s pair pooched +y and –y pz sy,ys EEWS-90.502-Goddard-L09

  48. Now reconsider Bonding H atom to all 3 states of C (2px)(2pz) Bring H1s along z axis to C and consider all 3 spatial states. C 2pz singly occupied. H1s can get bonding Get S= ½ state, Two degenerate states, denote as 2P (2py)(2pz) (2px)(2py) Now we can get a bond to the lobe orbital just as for BeH EEWS-90.502-Goddard-L09

  49. Is the 2P state actually 2P? The presence of the lobe orbitals might seem to complicate the symmetry Ψyz=A[(sx)(xs)+(xs)(sx)](ab-ba)(ya) (zH bond)2] Ψxz=A[(sy)(ys)+(ys)(sy)](ab-ba)(xa)(zH bond)2)] To see that there is no problem, rewrite in the CI form (and ignore the zH bond) Ψyz=A[(s2 – l x2)](ab-ba)(ya)] Now form a new wavefunction by adding - l y2 to Ψyz Φyz ≡A[s2 – l x2 – l y2](ab-ba)(ya)] But the 3rd term is A[y2](ab-ba)(ya)]= – l A[(ya)(yb)(ya)]=0 Thus Φyz = Ψyz and similarly Φxz = A[s2 – l x2 – l y2](ab-ba)(xa)] = Ψxz But the 2s terms [s2 – l x2 – l y2] are clearly symmetric about the z axis. Thus these wavefunctions have 2P symmetry EEWS-90.502-Goddard-L09

  50. Bonding of H to lobe orbital of C, Long R At large R the lobe orbitals of C are already hybridized Thus the wave function is A{(pxa)(pya)[(sz)(zs)+(zs)(sz)](ab-ba)(Ha)} Unpaired H 2s pair pooched +z and –z xy open shell Here the H overlaps slightly more with sz than with zs, but the spin on sz is half the time a Thus at large R we obtain a slightly repulsive interaction. EEWS-90.502-Goddard-L09

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