Is CH₂ linear or bent?

Wavefunctions for CH₂

Assuming that the two non-bonded valence electrons will go into the two unused valence orbitals on carbon we can write unnormalized wavefunctions for the triplet and singlet states of methylene. The triplet state has both electrons of the same spin. The Pauli Exclusion Principle allows only one way to place two electrons of the same spin (by convention they are both alpha spin) in two orbitals. There are three ways to obtain a singlet state: both the alpha and beta electrons in the red orbital, both electrons in the blue orbital, or a linear combination of one electron in each orbital. The relative importance of each "determinant" (electron configuration) to the overall wavefunction is dictated by the coefficients c1, c2, and c3. Hartree-Fock theory describes wavefunctions using only one determinant. For most molecules, there is only one important determinant but for singlet carbenes that is not the case. Complete active space self-consistent field (CASSCF) theory allows the user to determine how many electrons and orbitals are important to the question at hand. The wavefunctions shown below are those described by a CAS(2,2) calculation, where all ways of puting two electrons in two orbitals are considered. A more complete wavefunction would consider all of the valence electrons in all of the valence orbitals. For CH2 that would be a CAS(6,6).

Since RHF and CASSCF are variational, the lowest energy is closest to the true Schrödinger solution. Increasing the number of basis functions will give a better description of this molecule. We'll start with a very small basis set, 3-21G, and progress to a very large basis set, 6-311++G(3df,3pd). We will find out whether the computational description of methylene approaches the accepted values as the basis set and electron correlation are increased.

Procedures:

• Use ROHF/6-31G(d) (ROHF=restricted open shell hartree-fock) for the linear and bent triplet.
  • Note that ROHF is equivalent to CAS(2,2) since there is only one possible configuration.
  • Visualize the orbitals and copy down the optimized bond angle and energies.
• Use CAS(2,2)/6-31G(d) for the linear and bent singlet.
  • Check the "additional print" box under the General tab when the job is submitted to print the configuration weights in your log file.
  • Visualize the orbitals and copy down the optimized bond angle, configuration weights, and energies from the log file.

(Computation set 2: three calculations) Effect of electron correlation on bond angles and DE_ST

• Starting from the best geometry for triplet methylene, as determined above, optimize the triplet using full-valance CASSCF, CAS(6,6)/6-31G(d) and record its bond angle and energy.
• Using the same basis set and starting from the best geometry for singlet methylene, as determined above, optimize the singlet state without electron correlation (RHF) and, in a second calculation, using full-valance (6,6) CASSCF correlation. Record the bond angles and energies.
• Make a table like the one shown here to compare the absolute energies, bond angles, and DE_ST as a function of correlation.

  method	Singlet Absolute Energy	Triplet Absolute Energy	Singlet Bond Angle	Triplet Bond Angle	DEST
  RHF		from above			*
  CAS(2,2)	from above				*
  CAS(6,6)

  * Using ROHF triplet energy.

(Computation set 3: six calculations) Effect of basis set on bond angles and DE_ST

• Using the ROHF method for triplet methylene, reoptimize the geometry using each of the following basis sets: 3-21G, 6-31+G(d,p), and 6-311++G(3df,3pd)
• Using the CAS(2,2) method for singlet methylene, reoptimize the geometry using each of the following basis sets: 3-21G, 6-31+G(d,p), and 6-311++G(3df,3pd)
• Make a table like the one shown here to compare the absolute energies, bond angles, and DE_ST as a function of basis set. The number of basis functions is found in the log file.

  Basis Set	# of basis functions	Singlet Absolute Energy	Triplet Absolute Energy	Singlet Bond Angle	Triplet Bond Angle	DEST
  3-21G
  6-31G(d)	from above	from above	from above
  6-31+G(d,p)
  6-311++G(3df,3pd)

Analysis

Consider the following questions:

• Is methylene bent or linear? (answer for both electronic states)
• Use simplistic arguments (using Lewis structures and VSEPR, for instance) to argue the difference in bond angles for the two electronic states.
• Which electronic state is the ground state?
• Based on your configuration weights in the linear and bent ¹CH₂, what is the minimal electron correlation to properly determine the bond angle? That is, what is the minimal electron correlation to properly describe the ¹CH₂ wavefunction at all bond angles?
• Do the CAS(6,6) calculations show significant excitation out of the C-H sigma bonding orbitals or into the C-H anti-bonding orbitals? Do these excitations have an effect on the geometry or energy?
• How much does the energy change in going from RHF to CAS(2,2) in ¹CH₂? For both triplet and singlet methylene, how much more "correlation energy" is recovered in increasing the active space to full valence?
• According to your basis set table, what number of basis functions are required to converge on an answer for...
  • ³CH₂ energy
  • ³CH₂ geometry
  • ¹CH₂ energy
  • ¹CH₂ geometry
  • DE_ST
• The best experimental estimates are given below. Are the computational values converging towards these experimental values (plus or minus the reasonable experimental error limits) with infinite correlation and infinite basis sets?
  • ³CH₂ bond angle = 134
  • ¹CH₂ bond angle = 110
  • DE_ST = 9 kcal/mol