Поверхні потенціальної енергії основного та збуджених станів молекули ¹¹ВН у мультиреференсній теорії зв’язаних кластерів
У методi мультиреференсної теорiї зв’язаних кластерiв проведено розрахунки поверхнi потенцiальної енергiї (ППЕ) молекули ¹¹ВН в основному та збуджених станах. ППЕ апроксимовано за допомогою аналiтичних функцiй, що узагальнюють потенцiал Морзе. Дослiджено вплив точностi апроксимацiї ППЕ на розв’язок...
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Відділення фізики і астрономії НАН України
2010
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| Цитувати: | Поверхні потенціальної енергії основного та збуджених станів молекули ¹¹ВН у мультиреференсній теорії зв’язаних кластерів / Т.О. Кліменко, В.В. Іванов, Д.І. Лях // Український фізичний журнал. — 2010. — Т. 55, № 6. — С. 657-664. — Бібліогр.: 18 назв. — укр. |
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nasplib_isofts_kiev_ua-123456789-562132025-02-23T18:26:22Z Поверхні потенціальної енергії основного та збуджених станів молекули ¹¹ВН у мультиреференсній теорії зв’язаних кластерів Potential Energy Surfaces of the Ground and Excited States of ¹¹BH Molecule in the Multireference Coupled Cluster Theory Поверхности потенциальной энергии основного и возбужденных состояний молекулы ¹¹BH в мультиреференсной теории связанных кластеров Кліменко, Т.О. Іванов, В.В. Лях, Д.І. Атоми і молекули У методi мультиреференсної теорiї зв’язаних кластерiв проведено розрахунки поверхнi потенцiальної енергiї (ППЕ) молекули ¹¹ВН в основному та збуджених станах. ППЕ апроксимовано за допомогою аналiтичних функцiй, що узагальнюють потенцiал Морзе. Дослiджено вплив точностi апроксимацiї ППЕ на розв’язок радiального рiвняння Шредiнгера та значення спектроскопiчних параметрiв. The multireference state-specific coupled cluster theory is used for the calculation of the potential energy surfaces (PES) of ¹¹BH molecule in the ground and excited states. The PESs are approximated by a number of analytical functions generalizing the Morse potential. The solution of a radial Schr¨odinger equation and the values of spectroscopic constants depending on the accuracy of the PES approximation are analyzed. В методе мультиреференсной теории связанных кластеров для заданного состояния проведены расчеты поверхности потенциальной энергии (ППЭ) молекулы ¹¹ВН в основном и возбужденных состояниях. ППЭ аппроксимированы набором аналитических функций, обобщающих потенциал Морзе. Исследовано влияние точности аппроксимации ППЭ на решение радиального уравнения Шредингера и значения спектроскопических постоянных. Роботу виконано за пiдтримки фонду фундаментальних, прикладних i пошукових науково-дослiдницьких робiт Харкiвського нацiонального унiверситету iменi В.Н. Каразiна. 2010 Article Поверхні потенціальної енергії основного та збуджених станів молекули ¹¹ВН у мультиреференсній теорії зв’язаних кластерів / Т.О. Кліменко, В.В. Іванов, Д.І. Лях // Український фізичний журнал. — 2010. — Т. 55, № 6. — С. 657-664. — Бібліогр.: 18 назв. — укр. 2071-0194 PACS 31.15.bw https://nasplib.isofts.kiev.ua/handle/123456789/56213 539.194+544.182.5 uk Український фізичний журнал application/pdf application/pdf Відділення фізики і астрономії НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| topic |
Атоми і молекули Атоми і молекули |
| spellingShingle |
Атоми і молекули Атоми і молекули Кліменко, Т.О. Іванов, В.В. Лях, Д.І. Поверхні потенціальної енергії основного та збуджених станів молекули ¹¹ВН у мультиреференсній теорії зв’язаних кластерів Український фізичний журнал |
| description |
У методi мультиреференсної теорiї зв’язаних кластерiв проведено розрахунки поверхнi потенцiальної енергiї (ППЕ) молекули ¹¹ВН в основному та збуджених станах. ППЕ апроксимовано за допомогою аналiтичних функцiй, що узагальнюють потенцiал Морзе. Дослiджено вплив точностi апроксимацiї ППЕ на розв’язок радiального рiвняння Шредiнгера та значення спектроскопiчних параметрiв. |
| format |
Article |
| author |
Кліменко, Т.О. Іванов, В.В. Лях, Д.І. |
| author_facet |
Кліменко, Т.О. Іванов, В.В. Лях, Д.І. |
| author_sort |
Кліменко, Т.О. |
| title |
Поверхні потенціальної енергії основного та збуджених станів молекули ¹¹ВН у мультиреференсній теорії зв’язаних кластерів |
| title_short |
Поверхні потенціальної енергії основного та збуджених станів молекули ¹¹ВН у мультиреференсній теорії зв’язаних кластерів |
| title_full |
Поверхні потенціальної енергії основного та збуджених станів молекули ¹¹ВН у мультиреференсній теорії зв’язаних кластерів |
| title_fullStr |
Поверхні потенціальної енергії основного та збуджених станів молекули ¹¹ВН у мультиреференсній теорії зв’язаних кластерів |
| title_full_unstemmed |
Поверхні потенціальної енергії основного та збуджених станів молекули ¹¹ВН у мультиреференсній теорії зв’язаних кластерів |
| title_sort |
поверхні потенціальної енергії основного та збуджених станів молекули ¹¹вн у мультиреференсній теорії зв’язаних кластерів |
| publisher |
Відділення фізики і астрономії НАН України |
| publishDate |
2010 |
| topic_facet |
Атоми і молекули |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/56213 |
| citation_txt |
Поверхні потенціальної енергії основного та збуджених станів молекули ¹¹ВН у мультиреференсній теорії зв’язаних кластерів / Т.О. Кліменко, В.В. Іванов, Д.І. Лях // Український фізичний журнал. — 2010. — Т. 55, № 6. — С. 657-664. — Бібліогр.: 18 назв. — укр. |
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Український фізичний журнал |
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ATOMS AND MOLECULES
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 6 657
POTENTIAL ENERGY SURFACES OF THE GROUND
AND EXCITED STATES OF 11BH MOLECULE
IN THE MULTIREFERENCE COUPLED
CLUSTER THEORY
T.A. KLIMENKO, V.V. IVANOV, D.I. LYAKH
V.N. Karazin Kharkiv National University
(4, Svobody Ave., Kharkiv 61077, Ukraine; e-mail: vivanov@ univer. kharkov. ua )
PACS 31.15.bw
c©2010
The multireference state-specific coupled cluster theory is used for
the calculation of the potential energy surfaces (PES) of 11ВН
molecule in the ground and excited states. The PESs are approxi-
mated by a number of analytical functions generalizing the Morse
potential. The solution of a radial Schrödinger equation and the
values of spectroscopic constants depending on the accuracy of the
PES approximation are analyzed.
1. Introduction
Calculations of potential curves (or potential energy sur-
faces) and molecular spectroscopic constants (SC) still
remain rather a complicated problem of quantum chem-
istry. The main difficulties are related to the investi-
gation of PESs of electronically excited states, dissoci-
ation processes, and processes of formation and decay
of chemical bonds. An important peculiarity of these
structural-chemical situations consists in the presence
of a set of degenerate or quasidegenerate energy levels
and, rather often, the open-shell character of an electron
state. As a result, there arise the considerable correla-
tion effects that cannot be described, in principle, in the
framework of the single-determinant approximation. In
this connection, a special attention is paid to the de-
velopment of multiconfiguration methods involving the
electron correlation. At the present time, there exists
a number of computer programs realizing different ver-
sions of quantum-chemical methods in the framework of
the non-empirical (ab initio) methodology allowing one
to take both dynamic and non-dynamic components of
the electron correlation into account (in varying levels
of accuracy). A special interest is attracted by study-
ing the approaches based on the Coupled Cluster (СС)
theory.
During the recent decade, the CC method has ac-
quired a reputation as a high-precision technique involv-
ing the electron correlation in non-empirical calculations
of small- and medium-size molecular systems. The guar-
anteed size extensivity makes the CC theory an attrac-
tive instrument for the description of various physical-
chemical processes and effects (intermolecular interac-
tion, spectroscopy, nonlinear optical properties, and so
on). However, a considerable success of this theory is re-
lated, first of all, to the calculation of closed shells near
an equilibrium geometry. When describing the degen-
erate or quasi-degenerate chemical systems, the tradi-
tional CC method encounters a number of problems. It
is caused by the fact that, with increase in internuclear
distances or in the presence of quasidegeneracy effects,
the electron wave function of a molecular system can in-
clude several determinants with large weights. In these
situations, it is necessary to accurately consider the con-
tributions of some most important configurations. Thus,
the corresponding generalization of the CC theory rep-
resents an urgent problem.
The majority of the up-to-date quantum-chemical cal-
culations involving the electron correlation is based on
the Hartree–Fock determinant as an initial reference
state. This means that all excitations are generated rel-
ative to this Hartree–Fock determinant. An alternative
to this approach is presented by the method in the mul-
tidimensional model space (multireference, MR), where
a set of reference determinants is constructed. All nec-
essary excitations are generated relative to this set of
reference determinants. A comparative review of the
T.A. KLIMENKO, V.V. IVANOV, D.I. LYAKH
Fig. 1. CAS(2,2)CCSD/cc-pVDZ curves of the potential energy of
the ground and excited states of 11ВН molecule (zero line corre-
sponds to the dissociation limit of the ground state)
MR approaches can be found, for example, in [1]. It
is worth noting that the MRСС ideology as the most
general method of solving the correlation problem can
potentially solve any problem related to the dissociation
of a chemical bond.
A considerable part of the up-to-date MRCC calcula-
tions is based upon the so-called complete active space
(CAS). An active single-electron space is the minimal
set of molecular orbitals (or spin orbitals) that must be
taken into account in order to obtain a qualitatively cor-
rect solution of the investigated chemical problem. Using
active orbitals, one constructs a model (reference) space
– a set of determinants or their superpositions gener-
ated by various ways of the distribution of active or-
bitals by electrons. In the zero-order approximation of
the method, the electron correlation is taken into ac-
count with participation of only the group of “active”
orbitals.
In the given work, we proceed with testing our ear-
lier proposed multireference CC method using the active
orbital space (CAS Coupled Cluster Singles and Dou-
bles, CASCCSD) [2–4]. We realized this method as an
additional package of the well-known program complex
GAMESS [6]. Our previous calculations have demon-
strated a high accuracy of the CASCCSD method in
the description of the energy and the wave function as
compared to the accurate Full Configuration Interaction
(FCI) method. In this work, we perform calculations
of the PESs and SCs for three singlet states of 11ВН
molecule: the ground X1Σ+, the lowest totally symmet-
ric B1Σ+, and the lowest degenerate A1Π ones (Fig. 1).
We also study the influence of the accuracy of approx-
imation of the PESs on the SC values, as well as the
energies of the lowest vibrational terms with available
experimental values.
2. Theory
2.1. Multireference state-specific coupled cluster
theory
In the standard CC method, the reference state is pre-
sented by the only determinant constructed on the spin-
orbitals in the Hartree–Fock approximation. In the gen-
eral case, the wave function of the method has the fol-
lowing form:
|ΨCC〉 = exp(T̂1 + T̂2 + T̂3 + . . .+ T̂n)|0〉, (1)
where |0〉 is the reference state (Hartree–Fock determi-
nant), T̂1, . . . , T̂n are the operators of generation of a su-
perposition of excited determinants of the corresponding
multiplicity relative to the reference determinant. We
have
T̂1|0〉 =
∑
i,a
tai |ai 〉, (2)
T̂2|0〉 =
∑
i>j,a>b
tabij |abij 〉, (3)
T̂3|0〉 =
∑
i>j>k,a>b>c
tabcijk |abcijk〉, . . . , (4)
where tai , tabij , tabcijk , and so on stand for the amplitudes
characterizing the contribution of the corresponding con-
figurations (|ai 〉,|abij 〉,|abcijk〉, etc.). The indices i, j, k corre-
spond to occupied spin-orbitals, while a, b, c – to vacant
spin-orbitals in the reference state |0〉.
In practical applications of the CC theory, the sum
of the operators T̂ in expansion (1) is terminated at
some member resulting in different approximations of
the theory. The most widespread among them are the
theory explicitly taking only doubly excited configura-
tions into account (Coupled Cluster Doubles – CCD),
that explicitly taking singly and doubly excited config-
urations into account (Coupled Cluster Singles Doubles
– CCSD), that allowing for singly, doubly, and triply
excited configurations (Coupled Cluster Singles Doubles
and Triples – CCSDT), and so on. Each approximation
implies the corresponding expansion of the exponential
658 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 6
POTENTIAL ENERGY SURFACES OF THE GROUND AND EXCITED STATES
function into a series. For example, in the CCSD case
(T̂ = T̂1 + T̂2 ), one obtains
|ΨCCSD〉 = (1+T̂1+ T̂2+
1
2
T̂ 2
1 +
1
2
T̂ 2
2 + T̂1T̂2+ . . .)|0〉. (5)
Expansion (5) includes both linear and nonlinear
terms. On the one hand, this fact complicates the pro-
cedure of solving the Schrödinger equation, but, on the
other hand, these nonlinear terms provide the accuracy
and the additive energy separability of the method. The
working equations of the CCSD method explicitly deal
with amplitudes of only singly and doubly excited con-
figurations. However, due to the nonlinear components
of the wave function, the configurations of higher mul-
tiplicity are also approximately taken into account (em-
ulated). For example, the terms (T̂1T̂2 and T̂ 3
1 ) corre-
spond to the approximate description of triply excited
configurations. The component (T̂2T̂2) corresponds to
the approximate description of quadruply excited con-
figurations relative to the reference Hartree–Fock deter-
minant (|0〉). It is worth emphasizing once again that
the mentioned configurations of higher multiplicities are
taken into account only approximately, since they do not
correspond to definite amplitudes.
A direct generalization of (1) to the case of a many-
dimensional model space (multireference theory, MRCC)
yields the expression
|ΨMRCC〉 =
∑
I=1,M
exp(T̂ (I)
1 + T̂
(I)
2 + . . . )|ΦI〉, (6)
where the set of determinants |ΦI〉, I = 1,M forms a
reference space. The operators T̂ (I)
i in the MRCC the-
ory act on the corresponding selected reference determi-
nants. Real calculations are usually confined to double
excitations.
The choice of the reference space represents a key mo-
ment for the calculation of PESs of excited states. To
form the space, it is necessary to choose a set of active
orbitals. For this purpose, the complete set of molec-
ular orbitals is divided into core nonactive and valent
active ones (Fig. 2). The reference determinants are
those only, for which core orbitals are doubly occupied,
while active ones are populated by electrons in all pos-
sible ways. A reference space formed in such a manner
is complete (Complete Active Space, CAS). In the case
where two valent electrons are distributed among two
active orbitals (four spin-orbitals), the active space is
denoted as (2× 2) and has a dimension of 4.
In our calculations of the ground and excited states
of 11ВН molecule, the active space includes two orbitals
Fig. 2. Active space of orbitals of ВН molecule used for the calcu-
lation of the Σ+ states
with two distributed electrons. In this case, the descrip-
tion of the ground (X1Σ+) and excited (B1Σ+) states
requires the consideration of the bonding (3σ) and anti-
bonding (4σ) molecular orbitals. Four determinants ap-
pearing due to such a distribution of electrons are refer-
ence ones (Fig. 2). Calculating the excited (A1Π) states,
one should use the (3σ) and (1π) molecular orbitals as
active ones. Due to the symmetry, the distribution of
electrons among them generates only two determinants.
Our approach to the MRCC calculation (state-specific
method) consists in the choice of one of the reference
determinants as the so-called “formal reference” state.
Electron excitations relative to this determinant form
the rest of determinants of the reference state and the
corresponding excitations relative to the latter.
Thus, the wave function of our method (CASCCSD)
can be described in the following way (considering that
the formal reference determinant is |0〉):
|ΨCAS(2,2)CCSD〉 = exp(T̂ (ext)
1 + T̂
(ext)
2 )(1 + Ĉ1 + Ĉ2)|0〉.
(7)
Here, the operators Ĉ1 and Ĉ2 generate the electron dis-
tribution among the active orbitals (by forming super-
positions of reference determinants),
Ĉ1|0〉 =
∑
I,A
cAI |AI 〉, (8)
Ĉ2|0〉 =
∑
I>J,A>B
cAB
IJ |AB
IJ 〉, (9)
while the operators T̂1
(ext)
and T̂2
(ext)
form superposi-
tions of single and double excitations beyond the ref-
erence space relative to all reference determinants. In
our approach, these excitations represent excitations of
higher multiplicity with respect to |0〉. For example,
double excitations relative to the reference determinant
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 6 659
T.A. KLIMENKO, V.V. IVANOV, D.I. LYAKH
|AB
IJ 〉 are realized as quadruple ones with respect to the
“formal reference” determinant |0〉:
T̂
(ext)
2
(
cd
kl
)
|AB
I J 〉 = T̂4
(
ABcd
I J kl
)
|0〉. (10)
The practical choice of the “formal reference” determi-
nant depends on the type of the investigated specific
state. For example, in the case of calculating the 1Σ+
electron state, the reference function has the form
|0〉+ x|4σ4σ
3σ3σ
〉+ y
(
|4σ3σ〉+ |4σ3σ〉
)
, (11)
where the determinant |0〉 corresponds to a closed shell,
in which the 3σ molecular orbital is doubly occupied.
The rest of determinants of the reference function rep-
resent excitations with respect to |0〉. The coefficients
x and y can be found solving the Schrödinger equation.
Our calculations show that, when describing the ground
state (X1Σ+), one should take the |0〉 state as a “formal
reference” one, as this determinant makes the dominant
contribution to the wave function within the multirefer-
ence method of configuration interaction. When calcu-
lating the (B1Σ+) excited state, one can use one of the
singly excited determinants (|4σ3σ〉 or |4σ
3σ
〉).
The calculation of the 1Π state requires one to use the
following reference function:
|1π3σ〉+ |1π3σ〉. (12)
One of the determinants of this superposition (for exam-
ple, |1π3σ〉) can serve as a formal reference one. The second
determinant appears automatically, when the operator
Ĉ2 acts on |1π3σ〉.
As was already noted above, all excitations in our re-
alization of the multireference theory (CASCCSD) are
constructed relative to one determinant. Due to this
fact, it is necessary to take the chosen set of configu-
rations of higher multiplicity into account. The search
for the energies and the corresponding amplitudes of the
sought states follows the standard projection scheme of
the CC theory, according to which the Schrödinger equa-
tion is projected on the most important configurations,
whose contributions are characterized by definite ampli-
tudes:
〈0|H − ECAS(2,2)CCSD|ΨCAS(2,2)CCSD〉 = 0, (13)
〈ai |H − ECAS(2,2)CCSD|ΨCAS(2,2)CCSD〉 = 0, (14)
〈abij |H − ECAS(2,2)CCSD|ΨCAS(2,2)CCSD〉 = 0, . . . (15)
In these expressions, ECAS(2,2)CCSD stands for the to-
tal energy of the system. Our calculation scheme is de-
scribed in [2, 3], and [4] in detail.
2.2. Analytical form of the PES function
In order to describe a molecular state in the nonrela-
tivistic adiabatic approximation, it is necessary to find a
“realistic potential function” describing the dependence
of the energy on the internuclear distance V (r). In this
case, the form of the potential V (r) represents a source
of considerable errors in calculating SCs and energies
of vibrational-rotational terms. In our previous works
[6], we proposed a generalization of the Morse potential
(GM) in the following form:
VGM(r) =
q∑
m=2
αm[1− exp(−βm(r −Re))]m, (16)
where αm and βm are fitting parameters, Re is the equi-
librium internuclear distance. Function (16) at q = 2
(standard Morse potential) gives a qualitatively correct
form of the curve, but it is too rough for the accurate
approximation of data of ab initio calculations. One also
uses other functions generalizing the Morse potential, for
example, in the form of the Murrell–Sorbie (MS) poten-
tial [7]:
VMS(r) = −α
6≥q∑
m=1
βm(r−Re)m−1 exp(−β1(r−Re)), (17)
and the James–Coolidge–Vernon (JCV) potential [8]:
VJCV(r) =
q∑
m=2
αm[1− exp(−β(r −Re))]m. (18)
Here, α and β with the corresponding indices are the
fitting parameters.
Potentials (16)–(18) exponentially decrease at large
internuclear distances. However, it is known that, at
large distances, functions of the intermolecular (inter-
atomic) interaction take the form of power sums:
V (r →∞) ≈ α− C6
r6
− C8
r8
− . . . . (19)
In this expression, C6, C8, ... are the dispersion con-
stants determined by the nature of interacting atoms
(molecules). The generalized Lennard-Jones (LJ) func-
tion [9] satisfies this condition:
VLJ(r) = T dis − αy[2− y],
y =
(
Re
r
)n
(1 +
q∑
m=1
βm · zm),
660 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 6
POTENTIAL ENERGY SURFACES OF THE GROUND AND EXCITED STATES
z =
r/Re − 1
r/Re + 1
, (20)
where n = 6, while T dis corresponds to the dissociation
energy.
Searching for the parameters α and β with the help of
an iterative procedure, one minimizes the sum of squared
deviations of ab initio theoretical energies from the val-
ues obtained by the approximation in the whole PES
region (least square technique):
χ2 =
1
(N − p)
∑
i
[Ei − Vi]2, (21)
where Ei is the theoretical value of the energy
ECAS(2,2)CCSD at the point ri, Vi = Vi(ri) is the value of
the potential function (the result of approximation of the
set Ei) at the same point ri, N is the number of points
used in the approximation, and p is the total number of
the fitting parameters.
2.3. Calculation of molecular parameters
The complete description of a molecular state and the
calculation1 of a wide set of SCs require the solution of
the radial Schrödinger equation with the chosen poten-
tial V (r):
ϕ′′ν,J(r)− J(J + 1)ϕν,J(r)
r2
+2µ(Eν,J−V (r))ϕν,J(r) = 0,
(22)
where Eν,J denotes the eigenvalue corresponding to the
vibrational-rotational level, ϕν,J(r) is the wave function
of nuclear motion, ν is the vibrational quantum num-
ber, J is the rotational quantum number, and µ is the
reduced mass. The explicit form of the function V (r) al-
lows one to estimate the SCs of the investigated molecule
generally determined as the corresponding coefficients of
the Dunham expansion [10]:
Eν,J =
∑
`
∑
m
Y`m
(
ν +
1
2
)`
Jm(J + 1)m. (23)
The Dunham coefficients Y`m are expressed in terms of
different derivatives of the corresponding potential func-
tion V (r) (16)–(20) at the point r = Re calculated by
analytical or numerical differentiation. The error re-
lated to the number of levels in the molecular spectrum
taken into account when searching for the equilibrium
1 All calculations were performed in the atomic system of units.
vibrational frequency ωe and the anharmonicity ωexe
can vary between 20 cm−1 and 100 cm−1. The theo-
retical calculation of the SCs consists in the numerical
solution of the radial equation (22) with the potential
approximated in one of the possible ways [11]. Calcu-
lating the constants with the use of Eqs. (22) and (23),
it is necessary to take the experimental number of vi-
brational levels Eν,J into account. In this case, theo-
retical and experimental values can be compared. The
numerical differentiation of the total energy of the sys-
tem ECASCCSD “point-by-point” is the simplest way to
obtain the SCs. In this case, we used the standard for-
mulas of numerical differentiation by five points (see,
e.g., [12]):
F ′′5 =
1
12h2
[−E+2 + 16E+1 − 30E0 + 16E−1 − E−2],
F ′′′5 =
1
2h3
[E+2 − 2E+1 + 2E−1 − E−2],
F ′′′′5 =
1
h4
[E+2 − 4E+1 + 6E0 − 4E−1 + E−2]. (24)
Here, h is the step of differentiation (takes the values
from 0.01 a.u. to 0.1 a.u.), E0 corresponds to the energy
of the system at the minimum point Re, and E±n is the
energy of the system at the points Re ± nh.
It is worth noting that, generally speaking, the men-
tioned variants of calculating the SCs result in different
values of the constants.
3. Calculation Results
Table 1 presents the results of the CAS(2,2)CCSD cal-
culations of the SCs for three states of 11BH molecule.
The PESs are approximated by functions (16)–(20). We
also adduce the data of numerical differentiation. In
order to find the accurate values of the SCs, we stud-
ied the accuracy of approximation in the narrow re-
gion of the PES curve close to the minimum. In ad-
dition to the equilibrium vibrational frequency ωe and
the anharmonicity constant ωexe, Table 1 also presents
the centrifugal distortion constant D, the vibrational-
rotational interaction constant αe, and the rotational
constant Be. These parameters are found by numer-
ically integrating the energy ECAS(2,2)CCSD with the
use of 5 points. For comparison, we chose func-
tions (16), (18), and (20) with the same number of
the fitting parameters α and β. The data of Ta-
ble 1 exhibit some spread of the SC values caused by
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 6 661
T.A. KLIMENKO, V.V. IVANOV, D.I. LYAKH
Fig. 3. 11BH(X1Σ+), CAS(2,2)CCSD/cc-pVDZ, the accuracy of
approximation by the JCV potential, and the calculation error of
the vibrational terms
both the form of the potential and the mathematical
method of obtaining the corresponding derivatives (ana-
lytically for the nonlinear approximation or numerically
for the interpolation using five points). Nevertheless,
the values obtained in the approximate CAS(2,2)CCSD
method agree with the accurate FCI calculation for
the ground state. For the lowest excited states, the
CAS(2,2)CCSD calculation perfectly reproduces the val-
ues of equilibrium vibrational frequency and anhar-
monicity. However, a marked deviation from the experi-
mental value is observed for the vibrational-rotational
constants (αe). In this case, one should take into
account that the calculated value of αe depends on
many factors and particularly depends on the quality
of approximation of the potential function. This prob-
lem is studied in a number of works (see, e.g., [15])
stating among other things that none of the general-
ized Morse potentials can describe both most important
molecular constants αe and ωexe with the same accu-
racy.
Figure 3 presents the X1Σ+ PES curve and the ac-
curacy of approximation by potential (18) calculated as
the absolute deviation from the theoretical energies at
separate PES points ΔV . It is worth noting that such a
behavior of the approximation accuracy is typical of all
the investigated functions. In Fig. 3, one can see large
absolute deviations in the region of the curve bend reach-
ing 50 cm−1, which can introduce an additional error in
the calculation of the energy of vibrational terms.
The approximation of the A1Π PES in the whole
range of internuclear distances by functions (16)–(20)
yielded the dissociation energies DGM
e = 0.743 eV, DLJ
e =
0.766 eV, DJCV
e = 0.731 eV, and DMS
e = 0.743 eV
that agree with the experimental value Dexp
e =0.697
eV [16]. It is worth noting that the approximation
of ab initio data can considerably distort the accu-
racy of calculations reached by the quantum-chemical
method.
The quality of the description of excited states by the
CASCCSD method can be judged by the data of Table
2 displaying the results characterizing the A1Π PES: the
equilibrium internuclear distance Re, the position of the
barrier Rb, its height relative to the dissociation limit
V b, the dissociation energy De, the difference between
the PES minimum and maximum energies Vmax−min,
and the transition energy Te.
In addition, the accuracy of PES calculations depends
on the basis set of atomic orbitals. According to the data
of Table 2, the geometric PES parameters are systemat-
ically improved, when passing from the double to triple
split-valence polarized basis set cc-pVTZ.
T a b l e 1. CAS(2,2)CCSD and experimental values
[17] of the molecular parameters for the X1Σ+, A1Π, and
B1Σ+ states of 11ВН, cm−1
Function Re, Ȧ ωe ωexe D̄, 10−3 Be αe
Nαi,βi
= 6
X1Σ, R ∈ [0.9; 1.8], cc-pVDZ
LJ 1.2541 2343.5 49.26 1.140 11.61 0.4445
GM 1.2545 2346.8 50.75 1.134 11.60 0.4117
JCV 1.2548 2346.1 49.30 1.134 11.60 0.4010
FCI [18] 1.2559 2340.0 48.80 1.100 11.57 0.396
numerically
“5 points” 1.2550 2348.9 43.20 1.127 11.58 0.396
Exper. 1.2324 2366.9 49.39 1.242 12.02 0.412
A1Π, R ∈ [0.9; 2.0], cc-pVTZ
LJ 1.2176 2250.52 56.833 1.48 12.316 0.467
GM 1.2169 2251.37 57.392 1.48 12.330 0.498
JCV 1.2200 2250.40 56.732 1.46 12.267 0.430
Exper. 1.2195 2250.99 56.665 1.45 12.295 0.835
B1Σ+, R ∈ [0.9; 1.6], cc-pVDZ
GM 1.2309 2401.2 69.526 1.22 12.051 0.565
JCV 1.2073 2402.7 69.923 1.36 12.526 0.954
Exper. 1.2164 2399.9 69.51 1.26 12.339 0.485
662 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 6
POTENTIAL ENERGY SURFACES OF THE GROUND AND EXCITED STATES
The radial Schrödinger equation (22) was solved nu-
merically using a mesh of 500–1000 values of ϕν,J
in the interval r = [rmin, rmax]. Such a parti-
tion appears sufficient for the estimation of the en-
ergy of the lowest (to ν = 7) vibrational terms ac-
curate to within 0.01 cm−1. The corresponding ex-
perimental energies of the terms in the “zero” vi-
brational state G(ν) = Eν,0 were reproduced with
the use of the RKR technique [13], and their val-
ues for the first five levels (ν = 0 − 4) amount to
1171.1, 3440.4, 5614.2, 7694.7, and 9684.0 cm−1, respec-
tively.
Equation (22) was solved with the use of potentials
(16) and (18). In this case, the accuracies of the de-
scription of vibrational states are comparatively equal
provided that the number of fitting parameters is large
enough. However, it is worth noting that, at q = 12, the
number of variable parameters in the case of the Morse
function is equal to 24, whereas it is only 13 for the
James–Coolidge–Vernon function. The absolute devia-
tions of the calculated values of ΔG(ν) from the exper-
imental data for five first levels (ν = 0 − 4) are equal
to −9.0, −29.4, −49.9, −70.8, and −92.1 cm−1, respec-
tively.
The transition energy Te is usually calculated from
the observed transitions with no regard for Y00 in the
excited and ground states. According to the RKR
data, Te(B1Σ+) = 0.2384603 a.u. (52 336 cm−1) [14].
We calculated the transition energy as Te(B1Σ+) =
Emin(B1Σ+)− Emin(X1Σ+) = 58 256 cm−1. A consid-
erable error in the excitation energy can be due to the
drawback of the cc-pVDZ basis set. The correspond-
ing transition energy obtained in the cc-pVТZ basis set
Te(B1Σ+) = 55 830 cm−1. Moreover, it is worth not-
ing that the calculation error irregularly varies along
the curve, as the structure of the wave function fun-
damentally changes with increase in the interatomic dis-
tance.
The performed calculations have demonstrated a high
efficiency of the CASCCSD technique. An active space
T a b l e 2. Theoretical and experimental parameters
of the A1Π state of 11ВН molecule
Parameter CAS(2,2)CCSD Experiment [16, 17]
cc-pVDZ cc-pVТZ
Re (Ȧ) 1.2424 1.2144 1.2195
Rb (Ȧ) 1,952 2.052 2.143
V b (еV) 0.22 0.14 ≥ 0.11
De (еV) 0.415 0.738 0.697
Vmax−min (еV) 0.632 0.88 ≥ 0.81
Te (сm−1) 24 445 23 436 23 136
of dimension 2×2 appears sufficient for qualitatively re-
producing the PESs of the ground and excited states of
ВН (the existence of the potential barrier in the A1Π
state and the second minimum in the B1Σ+ state).
The SC values calculated as the corresponding coef-
ficients of the Dunham expansion are in good agree-
ment with the experimental data. The performed anal-
ysis of a number of analytical functions used for the
approximation of the PESs testifies to the sensitivity
of the calculated SC values to the form of the poten-
tial energy function and the number of fitting parame-
ters.
The work is supported by the Fund of fundamental,
applied, and scientific researches of the V.N. Karazin
Kharkiv National University.
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Received 27.07.09.
Translated from Ukrainian by H.G. Kalyuzhna
ПОВЕРХНI ПОТЕНЦIАЛЬНОЇ ЕНЕРГIЇ ОСНОВНОГО
ТА ЗБУДЖЕНИХ СТАНIВ МОЛЕКУЛИ 11BH
У МУЛЬТИРЕФЕРЕНСНIЙ ТЕОРIЇ
ЗВ’ЯЗАНИХ КЛАСТЕРIВ
Т.О. Клiменко, В.В. Iванов, Д.I. Лях
Р е з ю м е
У методi мультиреференсної теорiї зв’язаних кластерiв прове-
дено розрахунки поверхнi потенцiальної енергiї (ППЕ) молеку-
ли 11ВН в основному та збуджених станах. ППЕ апроксимова-
но за допомогою аналiтичних функцiй, що узагальнюють по-
тенцiал Морзе. Дослiджено вплив точностi апроксимацiї ППЕ
на розв’язок радiального рiвняння Шредiнгера та значення
спектроскопiчних параметрiв.
664 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 6
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