Small Be Clusters
- Original aim was to examine Be3 and Be4 with CCSD, and compare with
- Be has very large near-degeneracy correlation effects - suggest that
single-reference methods will be useless for small systems containing Be.
- Multireference treatments quickly get expensive, however,
- Be3 and Be4 are both totally symmetric singlets, the former equilateral
triangular and the latter tetrahedral.
Be3 Bond Lengths and Binding Energies
a Bond lengths in parentheses were not optimized.
|SCF||[4s 2p 1d]||(4.472)a||-3.2|
|CISD+Q||[4s 2p 1d]||(4.472)||8.9|
|CCSD||[4s 2p 1d]||4.472||4.2
|CPF||[4s 2p 1d]||4.488||8.6
|MCSCF||[4s 2p 1d]||(4.372)||-2.3
|MCRI||[4s 2p 1d]||4.373||14.2|
|SCF||[5s 3p 12d 1f]||(4.240)||-1.4
|CCSD||[5s 3p 12d 1f]||4.240||11.3
|CPF||[5s 3p 12d 1f]||4.202||14.9
|MCSCF||[5s 3p 12d 1f]||(4.199)||.2
|MCRI||[5s 3p 12d 1f]||4.199||22.4
- Be3 is not bound at either the SCF or CASSCF levels.
- Single-reference treatments are not in very good agreement with MRCI.
So it appears essential to describe the nondynamical correlation, just to
get the dynamical correlation contribution to binding correct.
- Very substantial basis set effect.
- A method like CCSD that is exact for the separated atom
limit performs worse that CISD + Q, because there is less cancellation of
Small Be Clusters
Be4 Binding Energies
a Bond length fixed at 3.9ao.
|SCF||[4s 2p 1d]||32.9
|CCSD||[4s 2p 1d]||42.9
|MCSCF||[4s 2p 1d]||34.9
|MCRI||[4s 2p 1d]||56.1
|SCF||[5s 3p 2d 1f]||40.0
|CCSD||[5s 3p 2d 1f]||63.5
|CASSCF||[5s 3p 2d 1f]||45.0
|MCRI||[5s 3p 2d 1f]||77.3
- Be4 is bound at the SCF and CASSCF levels.
- Again, there is a very large dynamical correlation contribution to the
- Estimate binding energies as 24+- kcal/mol for Be3 and 83+-3kcal/mol
Be3 and Be4
|CCSD||[5s 3p 2d 1f]||4.239||11.3
|CCSD(T)||[5s 3p 2d 1f]||4.2317||20.4
|CMCRI||[5s 3p 2d 1f]||4.2300||22.5
CCSD(T) performs remarkably well for these sytems!
|CCSD||[4s 2p 1d]||4.041||44.2
|CCSD(T)||[4s 2p 1d]||4.060||58.5
|MCRI||[4s 2p 1d]||4.054||59.2
|CCSD||[5s 3p 2d 1f]||43.900||63.5
|CCSD(T)||[5s 3p 2d 1f]||3.921||79.5
Small Be Clusters: CCSD(T) Calculations
- Generate a full quartic force field for each cluster, and obtain
Be3 and Be4 Vibrations(a)
|CCSD||[5s 3p 2d 1f]||433||407
|MCRI||[5s 3p 2d 1f]||490||427
|CCSD(T)||[5s 3p 2d 1f]||480||417
|CCSD(T)b||[5s 3p 2d 1f]||459||400
(a)Harmonic frequency unless otherwise indicated.
|CCSD||[4s 2p 1d]||597||445||534
|MCRI||[4s 2p 1d]||602||451||529
|CCSD(T)||[4s 2p 1d]||602||436||527
|CCSD||[5s 3p 2d 1f]||667||480||581
|CCSD(T)||[5s 3p 2d 1f]||662||469||571
|CCSD(T)b||[5s 3p 2d 1f]||639||455||682
- Basis set effects are large, but remaining effects should be small
- fundamentals accurate to within 20 cm -1, and probably to within 10.
- IR intensity of the Be3 e' mode is very small. The t2 mode in Be4
might be observable.
- Harmonic frequencies would be useless.
- Also treated Mg and Ca trimers and tetramers - same excellent agreement
between MRCI and CCSD(T).
Structure of C3+
- Is the ground state of C3+ linear or cyclic?
- Linear form is 2E+, cyclic form has C2v symmetry (2B2 state):
Jahn-Teller distorted equilateral triangle.
- Experiments were initially analyzed as indicating a cyclic ground
state, but it was pointed out that this was not unambiguous.
- Grev et al. used DZP basis and CI method: cyclic by 7
- Raghavachari, and Martin et al., TZ2Pf QCISD (T), cyclic by
only 2 or 3 kcal/mol.
- Very multiconfigurational.
- Grev et al. discovered an error in their calculations:
C3+ DZP Basis Results
Cyclic/ Linear Separation (kcal/mol)
- Very substantial nondynamical correlation effects. But dynamical
correlation contribution (relative to MCSCF) is very small.
- Best multireference values do not support Grev et al. directly
computed single-reference numbers.
C3+ Extended Basis Results
[5s 3p 2d 1f] Cyclic/Linear Separation (kcal/mol)
- Comparing with the DZP results, we would expect the best MRCI
result in this basis to be about 4.2 kcal/mol.
- Incompleteness of the one-particle basis set should be responsible for
most of the remaining deficiencies in this calculation.
Structure of C3+
- Scuseria: CCSD(T) calculations in a [5s 4p 3d 2f 1g] basis set show
that basis set extension could give another 2.5 kcal/mol.
- Hard to see any effect that could push the separation below 4.2 kcal/mol.
- Estimate 5.2 +1.5, -1.0 kcal/mol, taking account of basis set and
N-particle space effects on our best computed number.
- What is the most stable form?
- Experiments: only the ring structure is seen.
- Is the ring structure the most stable? Or is it simply too costly
(energetically) to form the fullerene from the ring?
- Large system, relatively speaking: hard to do much better than DZP
C20 relative energies (eV)
LDA optimized geometries
SCF optimized geometries