臺灣博碩士論文加值系統

The potential energy curve of the ground state of Mn$_2$ has been
studied using a systematic sequence of complete active spaces.
Deficiencies of the routinely used active space, built from atomic $4s$
and $3d$ orbitals, has been identified and discussed. It is shown that
an additional $\sigma_g$ orbital, originating from the atomic virtual
$4p_z$ orbitals, is essential for a proper description of static
correlation in the $^1\Sigma_{g}^{+}$ state of Mn$_2$. The calculated
spectroscopic parameters of the $^1\Sigma_{g}^{+}$ state agree well with
available experimental data. The calculated equilibrium bond lengths
are located between 3.24 and~3.50~{\AA}, the harmonic vibrational
frequencies, between 44 and~72~cm$^{-1}$, and the dissociation energies,
between 0.05 and~0.09~eV.

A detailed analysis of a severe intruder state problem in the multistate
multireference perturbation theory (MS-MRPT) calculations on the ground
state of manganese dimer is presented. An enormous number of detected
intruder states ($>$5000) do not permit finding even an approximate
shape of the $X^1\Sigma_{g}^{+}$ potential energy curve. The intruder
states are explicitly demonstrated to originate from quasidegeneracies
in the zeroth-order Hamiltonian spectrum. The electronic configurations
responsible for appearance of the quasidegeneracies are identified as
single and double excitations from the active orbitals to the external
orbitals. It is shown that the quasidegeneracy problem can be
completely eliminated using shift techniques despite of its severity.
The resultant curves are smooth and continuous. Unfortunately, strong
dependence of the spectroscopic parameters of the $X^1\Sigma_{g}^{+}$
state on the shift parameter is observed. This finding rises serious
controversies regarding validity of employing shift techniques for
solving the intruder state problem in multistate multireference
perturbation theory.

Prediction of a false ground state with popular variants of
multireference perturbation theory (CASPT2 and MRMP) is reported. The
failure occurs for a remarkably simple chemical system: the Sc$_2$
molecule. Reasons for the failure are discussed and appropriate
remedies are suggested. The presented finding has far-reaching
consequences for all the chemical community giving a serious warning on
the applicability of multireference perturbation theory in the presence
of intruder states.

A systematic investigation of low-lying states of Sc$_2$ using
multireference perturbation theory (NEVPT2 and NEVPT3) indicates that
the ground state of this system is $^5\Sigma_u^-$ with
$r_e=2.611$~{\AA}, $\omega_e=241.8$~cm$^{-1}$, and $D_e=1.78$~eV. This
state is closely followed by other low-lying states of Sc$_2$:
$^3\Sigma_u^-$, $^5\Delta_u$, $^3\Pi_g$, $^1\Pi_g$, and $^1\Sigma_u^-$.
Our energy ordering of the $^5\Sigma_u^-$ and $^3\Sigma_u^-$ states
confirms the recent MRCI results of Kalemos \textit{et al.}
[\textit{J.Chem.Phys.} \textbf{132}, 024309 (2010)] and is at variance
with the earlier DMC predictions of Matxain \textit{et al.}
[\textit{J.Chem.Phys.} \textbf{128}, 194315 (2008)]. An excellent
agreement between the second- and third-order NEVPT results and between
the computed and experimental values of $\omega_e$ (241.8 vs.
238.9~cm$^{-1}$) for the $^5\Sigma_u^-$ state suggests high accuracy of
our predictions.

The potential energy curve of the ground state of Mn$_2$ has been
studied using a systematic sequence of complete active spaces.
Deficiencies of the routinely used active space, built from atomic $4s$
and $3d$ orbitals, has been identified and discussed. It is shown that
an additional $\sigma_g$ orbital, originating from the atomic virtual
$4p_z$ orbitals, is essential for a proper description of static
correlation in the $^1\Sigma_{g}^{+}$ state of Mn$_2$. The calculated
spectroscopic parameters of the $^1\Sigma_{g}^{+}$ state agree well with
available experimental data. The calculated equilibrium bond lengths
are located between 3.24 and~3.50~{\AA}, the harmonic vibrational
frequencies, between 44 and~72~cm$^{-1}$, and the dissociation energies,
between 0.05 and~0.09~eV.

A detailed analysis of a severe intruder state problem in the multistate
multireference perturbation theory (MS-MRPT) calculations on the ground
state of manganese dimer is presented. An enormous number of detected
intruder states ($>$5000) do not permit finding even an approximate
shape of the $X^1\Sigma_{g}^{+}$ potential energy curve. The intruder
states are explicitly demonstrated to originate from quasidegeneracies
in the zeroth-order Hamiltonian spectrum. The electronic configurations
responsible for appearance of the quasidegeneracies are identified as
single and double excitations from the active orbitals to the external
orbitals. It is shown that the quasidegeneracy problem can be
completely eliminated using shift techniques despite of its severity.
The resultant curves are smooth and continuous. Unfortunately, strong
dependence of the spectroscopic parameters of the $X^1\Sigma_{g}^{+}$
state on the shift parameter is observed. This finding rises serious
controversies regarding validity of employing shift techniques for
solving the intruder state problem in multistate multireference
perturbation theory.

Prediction of a false ground state with popular variants of
multireference perturbation theory (CASPT2 and MRMP) is reported. The
failure occurs for a remarkably simple chemical system: the Sc$_2$
molecule. Reasons for the failure are discussed and appropriate
remedies are suggested. The presented finding has far-reaching
consequences for all the chemical community giving a serious warning on
the applicability of multireference perturbation theory in the presence
of intruder states.

A systematic investigation of low-lying states of Sc$_2$ using
multireference perturbation theory (NEVPT2 and NEVPT3) indicates that
the ground state of this system is $^5\Sigma_u^-$ with
$r_e=2.611$~{\AA}, $\omega_e=241.8$~cm$^{-1}$, and $D_e=1.78$~eV. This
state is closely followed by other low-lying states of Sc$_2$:
$^3\Sigma_u^-$, $^5\Delta_u$, $^3\Pi_g$, $^1\Pi_g$, and $^1\Sigma_u^-$.
Our energy ordering of the $^5\Sigma_u^-$ and $^3\Sigma_u^-$ states
confirms the recent MRCI results of Kalemos \textit{et al.}
[\textit{J.Chem.Phys.} \textbf{132}, 024309 (2010)] and is at variance
with the earlier DMC predictions of Matxain \textit{et al.}
[\textit{J.Chem.Phys.} \textbf{128}, 194315 (2008)]. An excellent
agreement between the second- and third-order NEVPT results and between
the computed and experimental values of $\omega_e$ (241.8 vs.
238.9~cm$^{-1}$) for the $^5\Sigma_u^-$ state suggests high accuracy of
our predictions.

1 General Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The multiconfiguration self-consistent field approach . . . . . . . . . . 2
1.3 Multireference perturbation theory . . . . . . . . . . . . . . . . . . . 3
1.3.1 Rayleigh-Schr¨odinger perturbation theory . . . . . . . . . . . 4
1.3.2 The choice of ˆH0 . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Intruder states in multireference perturbation theory . . . . . . . . . 8
2 Ground state of Mn2 revisited 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Choice of active space . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Bonding mechanism in Mn2 . . . . . . . . . . . . . . . . . . . 16
2.3.3 Spectroscopic constants . . . . . . . . . . . . . . . . . . . . . 20
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Intruder states in MRPT: Mn2 30
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.1 MCQDPT potential energy curves . . . . . . . . . . . . . . . . 34
3.3.2 CASPT2 potential energy curves . . . . . . . . . . . . . . . . 37
3.3.3 Analysis of intruder states in the MCQDPT calculations . . . 39
3.3.4 Failure of shift techniques . . . . . . . . . . . . . . . . . . . . 42
3.3.5 Alternative methods of removing intruder states . . . . . . . . 43
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Intruder states in MRPT: Sc2 59
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 The low-lying states of the scandium dimer 67
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.1 Atomic calculations . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.2 Low-lying states of Sc2 . . . . . . . . . . . . . . . . . . . . . . 75
5.3.3 NEVPT3 results for the 5−
u and 3−
u states . . . . . . . . . . 80
5.3.4 A comment on the applicability of reduced valence active spaces 81
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6 General conclusions 91
A Publications list 94
A.1 Publications included in this dissertation . . . . . . . . . . . . . . . . 94
A.2 Publications not included in this dissertation . . . . . . . . . . . . . . 94
B Auxiliary material for: Chapter 2 96
B.1 Deficiencies of the (12o,14e) active space . . . . . . . . . . . . . . . . 96
B.2 PES with the (13o,14e) active space . . . . . . . . . . . . . . . . . . . 101
B.3 Larger active spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
B.4 BSSE corrections and non-size consistency error . . . . . . . . . . . . 106
B.5 How short can be the bond in Mn2? . . . . . . . . . . . . . . . . . . . 110
B.6 ORMAS estimation of the full-valence CASSCF PES . . . . . . . . . 110
C Auxiliary material for Chapter 3 116
C.1 Note on deficiencies of H0 in MCQDPT . . . . . . . . . . . . . . . . . 116
C.2 Note on multistate MCQDPT calculations . . . . . . . . . . . . . . . 116
C.3 CASPT2 calculations with various values of the IPEA shift . . . . . . 118
C.4 Excluding terms from the perturbation . . . . . . . . . . . . . . . . . 119
D Auxiliary material for Chapter 4 121

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