Using Symmetry to Generate Molecular Orbital Diagrams review a few MO concepts generate MO for XH 2, H 2 O, SF 6 Formation of a bond occurs when electron density collects between the two bonded nuclei (ie., Ψ 2 MO = large in the region of space between nuclei) Antibonding Molecular Orbital Bonding Molecular Orbital Spin-pairing
. the valence electrons draw the atoms together until the core electrons start to repel one other, and that dominates 2
the total number of molecular orbitals (MO s) generated= total number of atomic orbitals ( s) combined Atom A (4 s) + Atom B (4 s) Molecule A B (8 MO s) in order to overlap... Atom A (4 s) + Atom B (4 s) s must have: similar energies identical symmetries Molecule A B (8 MO s) 3
Atom A Atom B σ bond doesn t change sign upon rotation about internuclear axis Atom A Atom B σ bond (antibond) σ since still doesn t change sign upon rotation about internuclear axis Atom A Atom B σ bond doesn t change sign upon rotation about internuclear axis Atom A Atom B σ bond (antibond) antibonding since changes sign upon rotation about axis perpendicular to internuclear axis 4
Atom A Atom B π bond changes sign upon rotation about internuclear axis Atom A Atom B π bond (antibond) π since still changes sign upon rotation about internuclear axis antibonding since changes sign upon rotation about axis perpendicular to internuclear axis 5
Additional orbital labels, used by your book, describe orbital symmetry with respect to inversion (x,y,z) i ( x, y, z) inversion center Additional orbital labels, used by your book, describe orbital symmetry with respect to inversion i g= gerade, if orbital does not change phase upon inversion π g * i u= ungerade, if orbital does change phase upon inversion π u 6
Additional orbital labels, used by your book, describe orbital symmetry with respect to inversion i σ g g= gerade, if orbital does not change phase upon inversion i σ u * u= ungerade, if orbital does change phase upon inversion the amount of stabilization and destabilization which results from orbital overlap depends on the type of orbital involved s orbital s orbital!e stab (s orbital) p orbital p orbital!e stab (p orbital) MO MO ΔΕ stab (s-orbital) > > ΔΕ stab (p-orbital) 7
the amount of stabilization and destabilization which results from orbital overlap depends on the type of orbital involved p orbital p orbital d orbital!e stab (p orbital) MO d orbital!e stab (d orbital) MO ΔΕ stab (p-orbital) > > ΔΕ stab (d-orbital) With p-orbitals need to consider two types of overlap, sigma (σ) and pi (π) p orbital p orbital "E stab (!) p orbital p orbital "E stab (#) # MO! MO ΔΕ stab (sigma) > > ΔΕ stab (pi) 8
Sigma overlap is better Δ stab (sigma) > > Δ stab (pi) One also needs to consider energy match!e stab (better E match)!e stab (poor E match) s are closer in energy s are further apart in energy ΔΕ stab (better E match) > > ΔΕ stab (poor E match) 9
Molecular Orbital Diagrams of more complicated molecules XH 2 (D h ) H 2 O (C 2v ) SF 6 (O h ) XH 2 (D h ) general MO diagram layout linear a linear combination of symmetry adapted atomic orbitals (LGO s) correlation lines central atom s (X s) atomic orbitals H X H molecular orbitals atomic orbitals of terminal atoms H + H 2 0
a linear combination of symmetry adapted atomic orbitals (LGO s)... Taken individually, the hydrogen atoms do not possess the symmetry properties of the XH 2 molecule. Taken as a group they do, however. what does this mean? H H 2 H + H 2 LGO (2) LGO () atomic orbitals of terminal atoms H + H 2 What are the symmetry properties of this molecule? There are an infinite number of these C 2 C2 H X H C...and an infinite number of mirror planes perpendicular to the C axis, which contain the C 2 s.
Taken individually, this hydrogen atom does not possess C 2 symmetry about axes that are perpendicular to the C. C 2 C2 H X C Thus, the individual hydrogen atoms do not conform to the molecular D h symmetry. s-orbital symmetry label XH 2 s point group symmetry label given to identify the p z orbital of atom X (central atom) atom X s p z atomic orbitals C 2 symmetry label given to identify the p x and p y orbitals of atom X (central atom) atom X s p x and p y atomic orbitals atom X s s orbital 2
We can perform symmetry operations on orbitals as well as molecules We can perform symmetry operations on orbitals as well as molecules the rows of numbers that follow each orbitals symmetry label tell us how that orbital behaves when operated upon by each symmetry element 3
C 2 We can perform symmetry operations on orbitals as well as molecules the rows of numbers that follow each orbitals symmetry label tell us how that orbital behaves when operated upon by each symmetry element a means that the orbital is unchanged by the symmetry operation a means that the orbital changes phase as a result of the symmetry operation a 0 means that the orbital changes in some other way as a result of the symmetry operation a cos θ occurs with degenerate sets of orbitals (eg. (p x, p x )) that take on partial character of one another upon performance of a symmetry operation 4
s-orbital C 2 C2 C rotation about the C, and C 2 axes leaves the s orbital unchanged C 2 thus the character in the row highlighted for the s-orbital, and under the columns headed C, and C 2 5
on the other hand, the p z orbital changes phase upon rotation about any of the C 2 axes C 2 pz-orbital C 2 rotation about the C 2 axis changes the phase of the pz orbital pz orbital C2 C rotation about the C axis leaves the pz orbital unchanged 6
but a + plus one under the column headed C C 2 thus the (negative one) character in the column headed C 2 When the px and py are rotated about the C axis, they move, rather than transform into themselves C 2 Thus the 0 in the column labeled 2C 7
inversion, on the other hand, causes both the px and py to change phase C 2 Thus the 2 in the column labeled i The 2 indicates that both orbitals transform as a set the orbitals are defined in terms of a fixed coordinate system p x -orbital x-axis C 2 C 2 Rotation about the C axis moves thep x orbital into a position where it no longer looks like a px orbital y-axis H X H z-axis C The z-axis is selected so that it either coincides with the highest-fold rotation axis, or the internuclear axis, or both For all C 2 s except those that coincide with the x- and y-axes, rotation about C 2 causes the p x orbital to take on some p y orbital character 8
Rotation about the C 2 s that do not coincide with the x- or y-axes, causes the p x orbital to take on some p y orbital character p x -orbital x-axis C 2 y-axis H X H z-axis C In fact, rotation about the C 2 axis that bisects the x- and y-axes converts the p x orbital into the p y orbital.this indicates that these two orbitals must be degenerate (ie have identical energies) C 2 Thus the cos φ term in the column labeled C 2 9
The number in the E column tells you how many orbitals transform together as a set inversion, on the other hand, causes both the px and py to change phase, but maintain their identities C 2 Thus the 2 in the column labeled i The 2 indicates that both orbitals transform as a set p x -orbital x-axis y-axis z-axis inversion center, i inversion thru i changes the phase of the p x orbital 20
Thus, we have determined the symmetry labels, generally referred to as Mulliken symbols,, for the s and p z, p x,and p y -orbitals on the central atom X p z XH 2 (D h ) a u (σ u ) p x p y e u (π u ) 2s a g (σ g ) This explains the labels shown for the central atom X on the left central atom s (X s) atomic orbitals 2
We determine the symmetry labels for the LGO s in the same manner, ie., by examining their symmetry with respect to the symmetry operations of the D h point group LGO (2) H H 2 H + H 2 LGO () LGO(2) C 2 rotation about the C 2 axes changes the phase of LGO(2) C2 C rotation about the C axis leaves LGO(2) unchanged 22
...thus it appears that LGO(2) possesses a u symmetry C 2... symmetry identical to that of the central atom X s p z orbital XH 2 (D h ) p y p x e u (π u ) p z a u (σ u ) 2s a g (σ g ) central atom s (X s) atomic orbitals..thus, the symmetry of LGO(2) matches that of the X atom s p z orbital..this interaction is symmetry allowed a u (σ u ) atomic orbitals of terminal atoms H + H 2 LGO (2) LGO ()..and LGO(2) will combine with the X p z orbital to form a new molecular orbital (MO) 23
LGO() C 2 rotation about the C 2 axes leaves LGO() unchanged C2 C rotation about the C axis leaves LGO() unchanged Or....thus it appears that LGO() could possess either a g, or a 2u symmetry C 2... a g would match orbitals on atom X, whereas a 2u would not. 24
if we examine the symmetry of LGO() with respect to both inversion, and reflection, we should be able to distinguish between these two options C 2 LGO() rotation about the C axes leaves LGO() unchanged C Reflection thru mirror plane Perpendicular to C leaves LGO() unchanged 25
thus, we can rule out a 2u as the symmetry assignment for LGO(), based on reflection, and improper rotation C 2 inversion leaves LGO() unchanged LGO() inversion center, i 26
thus, we can rule out a u, and a 2u as the symmetry assignment for LGO() C 2 leaving a g as the symmetry assignment for LGO() p z XH 2 (D h ) a u (σ u ) p x p y e u (π u )..thus, the symmetry of LGO() matches that of the X atom s s orbital a u (σ u ) LGO (2) 2s a g (σ g ) a g (σ g ) LGO () central atom s (X s) atomic orbitals atomic orbitals of terminal atoms H + H 2 27
XH 2 (D h )..inserting our correlation lines between orbitals of the same symmetry a * u (σ * u ) p z a u (σ u ) p x p y e u (π u ) a g * (σ g * ) e u (π u ) nonbonding a u (σ u ) LGO (2) 2s a g (σ g ) a u (σ u ) a g (σ g ) LGO () central atom s (X s) atomic orbitals a g (σ g ) atomic orbitals of terminal atoms H + H 2 28