"The wavefunction has an RHF -> UHF nstability" - Stability of Wavefunctions
最近用gaussian算的一个较大体系发现单重态和多重态的能量差就0.23ev,所以check了下波函数的稳定性,的确有点问题,提示"The wavefunction has an RHF -> UHF nstability.",google了下,发现了下面的网页对像我这样的新手还是比较有帮助的,故贴出来.[url=http://www.cup.uni-muenchen.de/ch/compchem/energy/stability.html]http://www.cup.uni-muenchen.de/ch/compchem/energy/stability.html[/url]
Wavefunctions generated by SCF calculations can be unstable in various ways:
1) The lowest energy wavefunction is a [color=#0000ff]singlet biradical[/color] instead of a closed shell singlet. A proper description of singlet biradicals at the Hartree-Fock level requires an UHF wavefunction. This is a typical case for an [color=#0000ff]RHF/UHF instability[/color]
2) A triplet state is energetically more favorable than the lowest lying singlet state.This will also lead to an [color=#0000ff]RHF/UHF instability[/color]
3) There is more than one solution to the SCF equations for the system and the calculationconverges to a less favorable one. This will lead to either a [color=#0000ff]RHF/RHF[/color]or [color=#0000ff]UHF/UHF instability[/color]
The various cases of wavefunction instability will be demonstrated using twosmall model systems, O2 and O3.
[b]1) molecular oxygen O2[/b]
Let us first calculate the Hartree-Fock energy for singlet dioxygen at the RHF/STO-5G level of theory (using the geometry optimized at this level) with the following input file:
[table][tr][td][color=#0000ff]#RHF/STO-5G scf=(tight,qc)
RHF/STO-3G singlet O2
0 1
O1
O2 1 r1
r1=1.22220
[/color][/td][td] [/td][td]O=O[/td][/tr][/table]
This yields the following result:
[color=#0000ff] SCF Done: E(RHF) = -148.886061396 a.u. after 4 cycles
Convg = 0.1803D-07 16 Fock formations.
S**2 = 0.0000 -V/T = 2.0033
**********************************************************************
Population analysis using the SCF density.
**********************************************************************
Orbital symmetries:
Occupied (SGG) (SGU) (SGG) (SGU) (PIU) (SGG) (PIU) (PIG)
Virtual (PIG) (SGU)
Unable to determine electronic state: partially filled degenerate orbitals.
[/color]Subsequent analysis of this wavefunction with the keyword combination[color=#0000ff]stable guess=read[/color]reads in the converged wavefunction and, by analysis of a number ofexcitations starting from the HF-solution, yields the following additional information:
[color=#0000ff] ***********************************************************************
Stability analysis using singles matrix:
***********************************************************************
Eigenvectors of the stability matrix:
Excited state symmetry could not be determined.
Eigenvector 1: Triplet-?Sym Eigenvalue=-0.2009390
7 -> 9 0.70536
Excited state symmetry could not be determined.
Eigenvector 2: Triplet-?Sym Eigenvalue=-0.1078684
8 -> 9 0.70711
Excited state symmetry could not be determined.
Eigenvector 3: Singlet-?Sym Eigenvalue= 0.0000000
8 -> 9 0.70711
Excited state symmetry could not be determined.
Eigenvector 4: Triplet-?Sym Eigenvalue= 0.0667651
6 -> 9 0.65552
7 -> 10 0.26170
Excited state symmetry could not be determined.
Eigenvector 5: Triplet-?Sym Eigenvalue= 0.2052332
5 -> 9 0.70711
Excited state symmetry could not be determined.
Eigenvector 6: Singlet-?Sym Eigenvalue= 0.2170234
6 -> 9 0.66826
7 -> 10 0.22006
The wavefunction has an RHF -> UHF instability.
[/color]
This analysis points to a triplet state wavefunction lower in energy than the currentsinglet state. The actual triplet wavefunction is, however, not calculated explicitly.In order to find the optimized triplet wavefunction, a second calculation must be performed.Using the same geometry as before (RHF/STO-5G), the calculation is performed in onerun together with the stability analysis of the triplet wavefunction:
[color=#0000ff]#UHF/STO-5G stable scf=(tight,qc)
UHF/STO-3G triplet O2
0 3
O1
O2 1 r1
r1=1.22220
[/color]
The total energy of -148.968737 Hartree obtained in this UHF/STO-5G calculation is lower by 217 kJ/mol than the one obtained for the singlet state at the RHF/STO-3G level!Despite this large improvement, the stability analysis reveals one more problemwith the wavefunction:
[color=#0000ff] The wavefunction has an internal instability.
[/color]
From the additional data given by the stability analysis it appears that the tripletstate optimized with the default guess does NOT converge to the triplet state of correct symmetry (and lowest energy). This can be solved either by an appropriate [url=http://www.cup.uni-muenchen.de/ch/compchem/energy/hf2.html]manipulation of the initial guess[/url]as discussed before or with the keyword [color=#0000ff]guess=mix[/color] originallydesigned to provide an asymmetric initial guess for calculations on singlet biradicals.In either case, a new triplet state of different symmetry is obtained at somewhat lowertotal energy of -148.9720434, 225.7 kJ/mol below the singlet state. Stability analysisof this wavefunction now confirms that:
[color=#0000ff] The wavefunction is stable under the perturbations considered.
[/color]
The same result can be obtained by running the stability calculation with [color=#0000ff]stable=opt[/color].Under these latter conditions the constraints imposed upon the wavefunction are reduced incrementallyuntil a stable wavefunction is obtained. The key feature of the most stable wavefunction obtained fortriplet oxygen by either [color=#0000ff]stable=opt[/color] or [color=#0000ff]guess=mix[/color]is the reduced symmetry of the two highest lying orbitals.
[b]2) ozone O3[/b]
Calculation of the singlet state RHF/6-31G(d) energy of ozone at its experimental geometry with the followinginput
[color=#0000ff]#RHF/6-31G(d) stable scf=(tight,qc)
RHF/STO-3G singlet O3
0 1
O1
O2 1 1.2227
O3 1 1.2227 2 114.0451
[/color]
and subsequent stability analysis yields the following result:
[color=#0000ff] SCF Done: E(RHF) = -224.258537798 a.u. after 8 cycles
The wavefunction has an RHF -> UHF instability.
[/color]
As we have seen in the first example of O2, this can immediately be solved byperforming an UHF/6-31G(d) calculation on the corresponding triplet state of ozone. In contrastto the first example, however, the total energy of the triplet state obtained in this way is higherthan that of the singlet state obtained initially. Stability analysis also reveals an internalinstability as encountered before due to convergence to a state of wrong symmetry:
[color=#0000ff] SCF Done: E(UHF) = -224.226611298 a.u. after 10 cycles
The wavefunction has an internal instability.
[/color]
In this situation one must consider the possibility of a singlet diradical which requiresthe use of an UHF wavefunction even for a singlet state. Generating a broken symmetryguess for the singlet wavefunction with [color=#0000ff]guess=mix[/color] currently appears to work well only with the INDO guess:
[color=#0000ff]#UHF/6-31G(d) guess=(INDO,mix) stable scf=(tight,qc)
UHF/STO-3G singlet O3
0 1
O1
O2 1 1.2227
O3 1 1.2227 2 114.0451
[/color]
As a result a biradical singlet state is obtained that is 180.3 kJ/mol more favorable than thesinglet state described by the RHF wavefunction before. In this last case the stability analysisdetects no further problems with the wavefunction:
[color=#0000ff]SCF Done: E(UHF) = -224.327207261 a.u. after 10 cycles
The wavefunction is stable under the perturbations considered.
[/color]
Please observe that a broken symmetry UHF singlet wavefunction is only obtainedusing the [color=#0000ff]guess=mix[/color] keyword. If this is not used, theinitial guess is chosen such that the SCF calculation converges to the RHF wavefunction evenwith the UHF Ansatz. An alternative strategy to obtain the energetically most favorablesinglet wavefunction for ozone involves the use of [color=#0000ff]stable=opt[/color]. [url]http://chem8.org/bbs/viewthread.php?tid=3016[/url]
:) :) 虽然不懂。
还是顶一下～
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