Applying Conventional Ab Initio and Density Functional Theory Approaches to Electric Property Calculations. Quantitative Aspects and Perspectives

Abstract

We have examined the predictive capability of density functional theory methods in calculations of electric polarizability and hyperpolarizability. We have focused on test cases belonging to three high-priority classes of molecular systems: “soft” metal clusters, novel types of compounds, and weakly bonded molecules. The performance of theoretical methods over arbitrary collections of molecular properties can be analyzed and classified by the introduction of a new methodology based on graph theoretic considerations and pattern recognition techniques.

Graphical Abstract

1 Introduction

The theory of electric polarizability is of fundamental importance to the rational approach and interpretation of large classes of phenomena [1]. In particular, these properties are of fundamental importance to intermolecular interaction studies [2], nonlinear optics [3], collision-induced spectroscopy [4], and the simulation of fluids [5, 6]. They are also routinely associated with general molecular characteristics as hardness [7], softness [8], hypersoftness [9], stiffness [10], and compressibility [11]. Understandably, polarizability is also linked to reactivity [12]. Another important field is QSAR, QSPR studies, and the understanding of pharmacological activity [13, 14].

In view of the important applications, the theoretical determination of electric properties of atoms, molecules, clusters, and even larger molecular architectures is rapidly expanding. The predictive capability of theoretical methods and convergence to the available experimental data has been closely examined in comprehensive reviews [15].

Two wide classes of theoretical methods are preferentially applied to the determination of electric polarizabilities: ab initio methods and density functional theory (DFT)-based approaches. Ab initio methods have been known to be converging reliably, displaying a high level of agreement to experimentally determined quantities. Very accurate ab initio treatments of electric polarizabilities are available for atoms and relatively small molecules. The distinct advantage of DFT methods lies in the possibility of economical, in a computational sense, treatments of relatively large molecular architectures. There, their advantage ends. It is usually very hard to determine the predictive capability of DFT-based methods or just to reasonably compare their performance to the presumably more accurate ab initio methodologies.

In this paper, we investigate the possibilities offered by widely used DFT methods. We have chosen test cases in three different, difficult classes of problems: (1) the linear and nonlinear polarizabilities of metal clusters, (2) the polarizabilities of novel compounds, and (3) the interaction-induced polarizability in weakly bonded systems. In particular, the three test cases are the sodium tetramer, a particularly “soft” molecule, the new compound HXeI, and the interaction polarizability of two water molecules in the dimer (H₂O)₂.

In previous work we employed an information theoretic approach to classify and systematically improve theoretical descriptions of molecules, introduced as arbitrary collections of data/properties [16, 17]. This approach relies on the availability of reference or accurate theoretical descriptions and is now easily accessible [18]. More recently, we have developed a more subtle, general approach based on graph theoretic arguments and pattern recognition techniques [19]. We rely on generalized metrics to introduce distance/proximity, order, and classification in spaces of theoretical descriptions. In addition, we introduce clustering in such spaces by the construction of a unique mathematical object, the minimum spanning tree (MST), and the performance of single-linkage cluster analysis (SLCA).

2 Theoretical Considerations

2.1 Electric Properties of Atoms, Molecules, and Clusters. Basic Theory and Computational Aspects

Our use of the theory of electric polarizability follows in all aspects, including the basic theoretical philosophy and terminology, the classic papers of Buckingham [20] and McLean and Yoshimine [21].

The energy (E ^p) and perturbed electric moments (μ _α ^p, Θ _αβ ^p, Ω _αβγ ^p) of an uncharged molecule in a weak, static electric field can be expanded as

$$ \begin{array}{llllll} {{E}^{\mathrm{p}}} \equiv {{E}^{\mathrm{p}}}({{F}_{\alpha }},\,{{F}_{{\alpha \beta }}},\,{{F}_{{\alpha \beta \gamma }}},\,{{F}_{{\alpha \beta \gamma \delta }}}, \ldots ) \\ = {{\mathbf{E}}^0} - {{{\boldsymbol{\mu}}}_{\alpha }}{{F}_{\alpha }} - (1/3){{{\boldsymbol{\varTheta}}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta}}}}}{{F}_{{\alpha \beta }}} - (1/15){{{\boldsymbol{\varOmega}}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma}}}}}{{F}_{{\alpha \beta \gamma }}} - (1/105){{{\boldsymbol{\varPhi}}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta}}}}}{{F}_{{\alpha \beta \gamma \delta }}} + \ldots \\ - (1/2){{{\boldsymbol{\alpha}}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta}}}}}{{F}_{\alpha }}{{F}_{\beta }} - (1/3){{{\mathbf{{\rm A}}}}_{{{\boldsymbol{\alpha}, \boldsymbol{\beta} \boldsymbol{\gamma}}}}}{{F}_{\alpha }}{{F}_{{\beta \gamma }}} - (1/6){{\mathbf{C}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta}, \boldsymbol{\gamma} \boldsymbol{\delta}}}}}{{F}_{{\alpha \beta }}}{{F}_{{\gamma \delta }}} \\ - (1/15){{{\mathbf{{\rm E}}}}_{{{\boldsymbol{\alpha}, \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta}}}}}{{F}_{\alpha }}{{F}_{{\beta \gamma \delta }}} + \ldots \\ - (1/6){{{\boldsymbol{\beta}}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma}}}}}{{F}_{\alpha }}{{F}_{\beta }}{{F}_{\gamma }} - (1/6){{{\mathbf{{\rm B}}}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta}, \boldsymbol{\gamma} \boldsymbol{\delta}}}}}{{F}_{\alpha }}{{F}_{\beta }}{{F}_{{\gamma \delta }}} + \ldots \\ - (1/24){{{\boldsymbol{\gamma}}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta}}}}}{{F}_{\alpha }}{{F}_{\beta }}{{F}_{\gamma }}{{F}_{\delta }} + \ldots, \\ \end{array} $$

(1)

$$ \begin{array}{llllll} \mu_{\alpha}^p = {{{\boldsymbol{\mu}}}_{{\boldsymbol{\alpha}}}} + {{{\boldsymbol{\alpha}}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta}}}}}{{F}_{\beta }} + (1/3){{\mathbf{A}}_{{{\boldsymbol{\alpha}, \boldsymbol{\beta} \boldsymbol{\gamma}}}}}{{F}_{{\beta \gamma }}} + (1/2){{{\boldsymbol{\beta}}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma}}}}}{{F}_{\beta }}{{F}_{\gamma }} + (1/3){{\mathbf{B}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta}, \boldsymbol{\gamma} \boldsymbol{\delta}}}}}{{F}_{\beta }}{{F}_{{\gamma \delta }}} \hfill \\ & + (1/6){{{\boldsymbol{\gamma}}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta}}}}}{{F}_{\beta }}{{F}_{\gamma }}{{F}_{\delta }} + \ldots, \hfill \\ \end{array} $$

(2)

$$ \Theta_{{\alpha \beta }}^p = {{\varTheta}_{{\alpha \beta }}} + {{\mathbf{A}}_{{{\boldsymbol{\gamma}, \boldsymbol{\alpha} \boldsymbol{\beta}}}}}{{E}_{\gamma }} + {{\mathbf{C}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta}, \boldsymbol{\gamma} \boldsymbol{\delta}}}}}{{F}_{{\gamma \delta }}} + (1/2){{\mathbf{B}}_{{{\boldsymbol{\gamma} \boldsymbol{\delta}, \boldsymbol{\alpha} \boldsymbol{\beta}}}}}{{F}_{\gamma }}{{F}_{\delta }} + \ldots, $$

(3)

$$ \Omega_{{\alpha \beta \gamma }}^p = {{{\boldsymbol{\varOmega}}}_{{{\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma}}}}} + {{\mathbf{E}}_{{{\boldsymbol{\delta}, \boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma}}}}}{{F}_{\delta }} + \ldots, $$

(4)

where the variables F _α , F _αβ , F _αβγ , etc., are the field, field gradient, etc. at the origin of the molecule. The terms in bold are the permanent properties of the system: energy (E ⁰), dipole (μ _α ), quadrupole (Θ _αβ ), octopole (Ω _αβγ ), and hexadecapole (Φ _αβγδ ) moment. The second-, third-, and fourth-order properties are the dipole polarizability (α _αβ ), dipole–quadrupole polarizability (A _α,βγ ), quadrupole polarizability (C _αβ,γδ ), dipole–octopole polarizability (E _α,βγδ ), first dipole hyperpolarizability (β _αβγ ), dipole–dipole–quadrupole hyperpolarizability (B _αβ,γδ ), and second dipole hyperpolarizability (γ _αβγδ ). The subscripts denote Cartesian components and a repeated subscript implies summation over x, y, and z. The number of independent components needed to specify the above tensors is strictly regulated by symmetry. In addition to the Cartesian components, of interest are the various invariants of some tensors. For the dipole polarizability and second dipole hyperpolarizability the mean is defined as

$$ \bar{\alpha } = (1/3){{\alpha}_{{\alpha \alpha }}} \ { {\rm and} } \ \bar{\gamma } = (1/5){{\gamma}_{{\alpha \alpha \beta \beta }}}. $$

(5)

For some systems of simple symmetry, as linear molecules, the invariants of dipole (hyper)polarizability tensors, mean values, and anisotropies are easily defined and commonly used. Most of them are measurable quantities or can be deduced from experimental observations.

$$ \begin{gathered} \bar{\alpha } = ({{\alpha}_{{zz}}} + 2{{\alpha}_{{xx}}})/3 \hfill \\ \varDelta \alpha = {{\alpha}_{{zz}}} - {{\alpha}_{{xx}}} \hfill \\ \bar{\beta } = (3/5)({{\beta}_{{zzz}}} + 2{{\beta}_{{zxx}}}) \hfill \\ \varDelta \beta = {{\beta}_{{zzz}}} - 3{{\beta}_{{zxx}}} \hfill \\ \bar{\gamma } = (3{{\gamma}_{{zzzz}}} + 8{{\gamma}_{{xxxx}}} + 12{{\gamma}_{{xxzz}}})/15 \hfill \\ {{\varDelta}_1}\gamma = 3{{\gamma}_{{zzzz}}} - 4{{\gamma}_{{xxxx}}} + 3{{\gamma}_{{xxzz}}} \hfill \\ {{\varDelta}_2}\gamma = {{\gamma}_{{zzzz}}} + {{\gamma}_{{xxxx}}} - 6{{\gamma}_{{xxzz}}}. \hfill \\ \end{gathered} $$

(6)

When sufficiently weak electric fields are applied it is possible to extract the electric properties of the molecule from the above expansions. In previous work, we have applied various computational schemes based on the finite-field [22] approach to the calculation of electric properties from perturbed atomic/molecular energies and induced multipole moments [23–28].

Various computational aspects of the theoretical determination of electric properties are available in books [29, 30] or comprehensive collections [31–33].

All calculations reported in the following sections have been performed with quantum chemical methods easily accessible via the widely used GAUSSIAN suite of programs. See GAUSSIAN 98 [34] and GAUSSIAN 03 [35]. This arsenal includes conventional ab initio methods and density functional theory (DFT) approaches. We will not give further details about the structure and predictive capability of these methods here. For the interested reader, extensive presentations of the above classes of methods are clearly presented in standard references [36–39].

The ab initio methods used are:

• SCF, self-consistent-field

• MP2, MP3, and MP4: second-, third-, and fourth-order Møller–Plesset perturbation theory

• DQ-MP4 and SDQ-MP4, partial fourth-order Møller–Plesset

• CCSD, singles and doubles coupled cluster

• CCSD(T), which includes an estimate of connected triples via a perturbational treatment. This is the method with the, presumably, highest predictive potential

The DFT methods include the widely used B3LYP, B3PW91, and mPW1PW91 and many more that have been occasionally employed in electric property calculations.

Basis sets in modern quantum chemistry is too broad a subject to be examined in detail here [40]. The search for Gaussian basis sets suitable for molecular property calculations is vital to computational quantum chemistry [41]. Many computational schools have reported significant work in the direction of basis set construction [42–44] and testing of their performance. The construction of purpose-oriented basis sets has attracted particular attention [45–49]. Early work by Dykstra and coworkers [50, 51] or Spackman [52] has elucidated many important aspects of the performance of basis sets in electric property calculations. In our computational efforts we favor the use of molecule-specific, purpose-oriented basis sets. The construction of molecule-specific, purpose-oriented basis sets for large molecular architectures or low-symmetry polyatomics is largely impractical. Nevertheless, previous work shows that for systems of reasonable size as atoms [23, 53], atomic anions or cations [54, 55], diatomics [56, 57] and triatomics [58–61], symmetric polyatomics [62–64], and clusters [65–68], one can easily control the construction of the basis set in order to obtain suitable basis sets for electric property calculations.

2.2 Interaction-Induced Electric Properties

Interaction electric properties, as dipole moment, polarizability, and hyperpolarizability are of fundamental importance to the analysis and interpretation of measurements and observations in collision- and interaction-induced spectroscopies [69, 70]. Considerable progress has been recorded in recent years either on theoretical issues or computational advances. We single out a few significant papers related to the above field. Głaz et al. [71] reported a study of the collision-induced hyper-Rayleigh light-scattering spectra of He–Ne atomic pairs. Chrysos et al. [72] reported a study of the CO₂–Ar collision-induced ν ₃ CO₂ band. A study of fundamental importance for collision-induced spectroscopy of gaseous CO₂, the determination of the exact low-order classical moments in collision-induced bands of linear rotors, was reported by Chrysos et al. [73]. A new treatment of the collision-induced Raman scattering by Ne–Ne was reported by Chrysos et al. [74]. Baranowska et al. [75] reported a theoretical study of the interaction-induced dipole moment and polarizability of CO–Ne. El-Kader et al. [76] determined the contributions of multipolar polarizabilities to the isotropic and anisotropic light scattering induced by intermolecular interactions in gaseous CH₄. Zvereva-Loëte et al. [77] and Buldakov et al. [78] reported an extensive study on the dipole moment surface and dipole polarizability surface for the CH₄–N₂ complex, a system of importance for the atmospheric physics of Titan. The calculation of the interaction-induced dipole moment, polarizability, and first and second hyperpolarizability of the H₂O–Rg (Rg = He, Ne, Ar, Kr and Xe) complexes was reported by Haskopoulos and Maroulis [79]. Hartmann et al. [80] reported an investigation of the far-infrared collision-induced absorption band in gaseous CO₂.

Our work on the interaction-induced polarizability and hyperpolarizability of the water dimer relies on the conventional supermolecule approach. Details on this approach have been given in previous work on interaction-induced electric properties. In this approach, the interaction-induced properties of the $\rm A\cdots B$ supermolecule are obtained as

$$ {{{\rm P}}_{{\rm int}}}({{\rm A}} \cdots {{\rm B}}) = {{\rm P}}({{\rm A}} \cdots {{\rm B}}) - {{\rm P}}({{\rm A}}) - {{\rm P}}({{\rm B}}). $$

(7)

In practice, the above equation is nearly exact when very large, nearly saturated basis sets are used. When truncated basis sets are used the basis set superposition error is removed by the counterpoise-correction method of Boys and Bernardi [81]. Thus, the above equation is replaced by

$$ {{{\rm P}}_{{\rm int}}}({{\rm A}} \cdots {{\rm B}}) = {{\rm P}}({{\rm A}} \cdots {{\rm B}}) - {{\rm P}}({{\rm A}} \cdots {{\rm X}}) - {{\rm P}}({{\rm X}} \cdots {{\rm B}}), $$

(8)

where P($\rm A\cdots X$) denote calculation of the property for subsystem A in the presence of the ghost orbitals of subsystem B and P($\rm X\cdots B$) for subsystem B in the presence of the ghost orbitals of subsystem A. It is easily shown that as the flexibility of the basis set increases one approaches a nearly ideal situation where P($\rm A\cdots X$) ≈ P(A) and P($\rm X\cdots B$) ≈ P(B).

For all computational aspects of the interaction-induced (hyper)polarizability of the water dimer, we lean heavily on previous experience of systems as CO₂–Rg [82], Ne–Ar [83], Xe–Xe [84], Kr–Xe [85], He–Ar [86], H₂–Ar [87], H₂–Ne [88], Kr–He [89], Kr–Ne [90], and H₂–Ne [91].

2.3 Proximity, Similarity, and Order in Spaces of Theoretical Descriptions

The evaluation of the performance of theoretical methods in atomic/molecular property calculations has been recognized as a formidable problem early enough. The quantification of the relative merit of theoretical methods is an essential part of modern computational quantum chemistry. Sometime ago we presented a methodology that relies on graph theoretic arguments and pattern recognition techniques to introduce order, classification, and clustering in spaces of arbitrary theoretical descriptions of atomic/molecular systems [92]. Our theory uses metric considerations to define distance/proximity and similarity in such spaces. The utility of such a methodology has been brought forth in various applications. Pattern recognition has long found application in Chemistry [93]. Our use of such techniques extends the application of pattern recognition (PR) to a hitherto unreached field: computational quantum chemistry (CQC). The logical strength of the analogy is made obvious by the following diagram:

Pattern recognition (PR)		Computational quantum chemistry (CQC)
Object	↔	Method
Features	↔	Molecular property values
Pattern	↔	Theoretical description (TD)
Pattern space	↔	Space of all TD

Thus in PR one has objects and in CQC methods. Objects are characterized by features and methods by molecular property values. A collection of features is a pattern; a collection of molecular property values a theoretical description (TD). A collection of patterns forms the pattern space, a collection of theoretical descriptions the space of all theoretical descriptions.

The above-mentioned methodology has found application in various quantum chemistry problems [94–96]. We give here only a few essential points and definitions.

Let TD _i be a collection of properties Q _mα where the index m denotes methods and the index α denotes properties. The two indices take values in the index sets I _α and I _m (respectively).

$$ {{\rm T}}{{{\rm D}}_i} = \{ {{Q}_{{{{\rm m}\upalpha}}}},m \in {{I}_{{\rm m}}},\alpha \in {{I}_{{\upalpha}}}\}. $$

(9)

We denote by TD the space of all theoretical descriptions TD _i :

$$ {{\rm TD}} = \left\{ {{{\rm T}}{{{\rm D}}_1},{{\rm T}}{{{\rm D}}_2}, \ldots {{\rm T}}{{{\rm D}}_N}} \right\}\quad {{\rm where}}\quad i = 1,2, \ldots N. $$

(10)

We define a generalized distance in the space of all TD by using the Minkowski metric. The distance between two theoretical descriptions TD _i and TD _j is defined as

$$ {{D}_{{ij}}} \equiv D\left( {{{\rm T}}{{{\rm D}}_i},{{\rm T}}{{{\rm D}}_j}} \right) = {{\left( {\sum\limits_{\alpha } {\frac{{{{{\left| {{{Q}_{{i\alpha }}} - {{Q}_{{j\alpha }}}} \right|}}^p}}}{{{{{\left( {\mathop{{{\max }}}\limits_{{ij}} \left| {{{Q}_{{i\alpha }}} - {{Q}_{{j\alpha }}}} \right|} \right)}}^p}}}} } \right)}^{{1/p}}},p \ge 1,1 \le i,j \le N. $$

(11)

In most applications we have used the Euclidean metric, that is p = 2.

Similarity between two theoretical descriptions TD _i and TD _j is then defined on the basis of distance/proximity as

$$ {{S}_{{ij}}} = 1 - \frac{{{{D}_{{ij}}}}}{{\mathop{{\max }}\limits_{{ij}} {{D}_{{ij}}}}},\quad 1 \leq i,j \le N. $$

(12)

By definition, $0 \le {{S}_{{ij}}} \le 1$.

To make a connection with graph theory we need a few standard definitions and interpretations. A reliable source of basic graph theory is the work of Chartrand and Lesniak [97]. A graph G is a finite nonempty set of objects called vertices together with a set of unordered pairs of vertices called edges. The vertex set of G is denoted V(G) and the edge set E(G). The cardinality p of V(G) is called the order of G and the cardinality q of E(G) the size of G. Let us consider the Cartesian product TD × TD. The graph of theoretical descriptions G _TD has as vertex set V(G _TD) the set of theoretical descriptions TD _i . The edge set E(G _TD) is a subset of above defined Cartesian product, E(G _TD) ⊂ TD × TD.

We assign to each edge of G _TD a real number, a weight. Consider the edge defined by TD _i and TD _j . We assign to the edge {TD _i , TD _j } the real number ${{D}_{{ij}}} \equiv D({{\rm T}}{{{\rm D}}_i},{{\rm T}}{{{\rm D}}_j})$. G _TD is now a weighted graph.

The diameter Diam G _TD of the graph of TD is defined as

$$ {{\rm Diam}}\quad {{G}_{{\rm TD}}} = {{\rm ma}}{{{\rm x}}_{{i,j \in V\left( {{{G}_{{\rm TD}}}} \right)}}}{{D}_{{ij}}} \equiv {{\rm ma}}{{{\rm x}}_{{{{\rm T}}{{{\rm D}}_i},{{\rm T}}{{{\rm D}}_j} \in V\left( {{{G}_{{\rm TD}}}} \right)}}}D\left( {{{\rm T}}{{{\rm D}}_i},{{\rm T}}{{{\rm D}}_j}} \right). $$

(13)

In view of the above, the definition of the similarity can now be written as

$$ {{S}_{{ij}}} \equiv S\left( {{{\rm T}}{{{\rm D}}_i},{{\rm T}}{{{\rm D}}_j}} \right) = 1 - \frac{{D\left( {{{\rm T}}{{{\rm D}}_i}, {{\rm T}}{{{\rm D}}_j}} \right)}}{{{{\rm Diam}}\quad {{G}_{{\rm TD}}}}},{{\rm T}}{{{\rm D}}_i},{{\rm T}}{{{\rm D}}_j} \in V\left( {{{G}_{{\rm TD}}}} \right). $$

(14)

Two more definitions are particular useful at this stage. The distance of fixed vertex u from a subset of the vertex set S ⊆ V(G _TD) is defined as

$$ d(u,S) = {{\rm mi}}{{{\rm n}}_{{x \in S}}}\{ d(u,x)\}. $$

(15)

A point of major importance to our methodology is the definition of a minimum spanning tree (MST). A spanning subgraph H of a graph G has vertex and edge sets V(H) ⊆ V(G) and E(H) ⊆ E(G) and is of the same order as G. A graph G of order p and size q is a tree if and only if it is acyclic and p = q + 1. The weight of a spanning tree in a connected graph is the sum of the weights of its edges. Thus a minimum spanning tree of G is a spanning tree of G of minimum weight.

Suitable algorithms exist for the construction of a MST [98].

Here is now a synoptic form of our methodology step by step:

• 1. Define the space of theoretical descriptions TD.

• 2. Calculate all distances between vertices or TD _i of the graph G _TD.

• 3. Calculate the distance and similarity matrix.

• 4. Construct a weighted minimum spanning tree MST using known algorithms.

• 5. Perform a single linkage cluster analysis (SLCA) by removing all edges from the MST characterized by weights above a given threshold D _T .

This analysis creates a partition of the MST in distinct clusters C ₁, C ₂, …, C _K . The union of all clusters is the vertex set of the G _TD graph: $V\left( {{{G}_{{\rm TD}}}} \right) = \mathop{{\bigcup {{{C}_i}} }}\limits_{{1 \leq i \le K}}$.

Last, it is important to introduce a distance between clusters for our analysis. The nearest neighbor distance between clusters C _m and C _n is defined as

$$ {{D}_{{nn}}}({{C}_m},{{C}_n}) = \mathop{{{\min }}}\limits_{{{{\rm T}}{{{\rm D}}_i} \in {{C}_m},{{\rm T}}{{{\rm D}}_j} \in {{C}_n}}} D({{\rm T}}{{{\rm D}}_i},{{\rm T}}{{{\rm D}}_j}). $$

(16)

3 Results

3.1 Sodium Tetramer, a Very Soft Molecule

The electric dipole polarizability of sodium clusters has been extensively studied, both experimentally [99–101] and theoretically [102–104]. In addition, the polarizability of the sodium atom is accurately known. The latest experimental value is α(Na) = 162.7 ± 0.8, reported by Ekstroem et al. [105]. Reliable theoretical values have been reported both for the Hartree-Fock limit and higher levels of theory. The most accurate theoretical value α(Na) = 162.88 ± 0.6 has been reported by Thakkar and Lupinetti [106]. Consequently, reference values are readily available for the differential polarizability (DP) and the differential-per-atom-polarizability (DPA), defined as [66, 107]:

$$ {{\rm DP}} \equiv {{\bar{\alpha }}_{{\rm diff}}} = \left( {\bar{\alpha }({{\rm N}}{{{\rm a}}_n}) - n\alpha ({{\rm Na}})} \right), $$

$$ {{\rm DPA}} \equiv {{\bar{\alpha }}_{{\rm diff}}}/n = \left( {\bar{\alpha }({{\rm N}}{{{\rm a}}_n}) - n\alpha ({{\rm Na}})} \right)/n = \left( {\bar{\alpha }({{\rm N}}{{{\rm a}}_n})} \right)/n - \alpha ({{\rm Na}}). $$

The respective hyperpolarizability quantities, differential hyperpolarizability (DH) and the differential-per-atom-hyperpolarizability (DHPA), are defined as

$$ {{\rm DH}} \equiv {{\bar{\gamma }}_{{\rm diff}}} = \left( {\bar{\gamma }({{\rm N}}{{{\rm a}}_n}) - n\gamma ({{\rm Na}})} \right), $$

$$ {{\rm DHPA}} \equiv {{\bar{\gamma }}_{{\rm diff}}}/n = \left( {\bar{\gamma }({{\rm N}}{{{\rm a}}_n}) - n\gamma ({{\rm Na}})} \right)/n = \left( {\bar{\gamma }({{\rm N}}{{{\rm a}}_n})} \right)/n - \gamma ({{\rm Na}}). $$

DH and DHPA are even more basis set and method sensitive than DP and DPA. As it has been shown elsewhere the above quantities are very useful as rigorous criteria for the analysis of the performance of quantum chemical methods [108, 109].

We have chosen as testing ground for our theoretical treatment the sodium tetramer (Na₄). Most of the data pertaining to Na₄ and used in this section have been published elsewhere [110]. The molecular geometry of the tetramer is a rhombus with side 3.64 Å and a short diagonal of 3.30 Å. The molecule is placed on the xz plane with the four Na atoms at (0,0,±1.65) and (±3.244549,0,0). All electric property calculations have been performed with a purpose-oriented molecule-specific basis set of [7s5p2d] type. The theoretical methods used are conventional ab initio and a selection of DFT approaches. The ab initio methods are SCF, MP2, MP3, DQ-MP4, SDQ-MP4, MP4, CCSD, and CCSD(T). The choice of DFT methods leans heavily on previous experience [111–115]: B3LYP, B3PW91, PBEPBE, PBEPW91, mPW1PW91, and mPW1PBE as implemented in the GAUSSIAN suite of codes.

The independent Cartesian components of the polarizability (α _αβ ) and hyperpolarizability (γ _αβγδ ) are given in Table 1. The method-dependence of the mean and the anisotropy of the dipole polarizability and the mean second hyperpolarizability are shown in Figs. 1, 2, and 3, respectively. Figure 1 shows clearly that electron correlation reduces the size of the mean dipole polarizability. The sequence $\bar{\alpha }({{\rm SCF}}) < \bar{\alpha }({{\rm MP}}2) < \bar{\alpha }({{\rm MP}}3) < \bar{\alpha }({{\rm MP}}4)$ shows a monotonic decrease of the values calculated with Møller–Plesset perturbation theory. Our best theoretical value $\bar{\alpha }({{\rm CCSD}({\rm T})})$ is quite close to $\bar{\alpha }({{\rm MP}}2)$ and $\bar{\alpha }({{\rm MP}}3)$. For comparison, we show on the same figure the DFT values for the mean calculated with mPW1PW91 and B3PW91. The former is close to the MP4 value and the latter close to SDQ-MP4. Figure 2 shows that the method-dependence of the dipole polarizability anisotropy values bear close resemblance to that of the mean. We observe again the monotonic Δα(SCF) < Δα (MP2) < Δα (MP3) < Δα (MP4). Agreement with the DFT methods is less obvious in this case. Δα(B3LYP) is quite close to Δα(MP2) but Δα(B3PW91) is significantly higher than the ab initio values. Inspecting Fig. 3, we are surprised to find that the various orders of Møller–Plesset perturbation theory display once more a monotonic behavior for the mean second dipole hyperpolarizability: $\bar{\gamma }({{\rm SCF}}) < \bar{\gamma }({{\rm MP}}2) < \bar{\gamma }({{\rm MP}}3) < \bar{\gamma }({{\rm MP}}4)$. $\bar{\gamma }({{\rm MP}}3)$ is quite close to our, presumably, most accurate $\bar{\gamma }$ (CCSD) and $\bar{\gamma }$ (CCSD(T)). What is more three conventional DFT methods as B3LYP, B3PW91 and mPW1PW91 yield mean second hyperpolarizability values close enough to the high-level ab initio ones.

Table 1 Static (hyper)polarizability of Na₄ calculated with ab initio and DFT methods (10⁻³ × γ _αβγδ ). All quantities in atomic units

Fig. 1

Method dependence of the mean dipole polarizability of Na₄

Fig. 2

Method dependence of the anisotropy the dipole polarizability of Na₄

Fig. 3

Method dependence of the mean second hyperpolarizability of Na₄

We have used the data in Table 1 to calculate distance/proximity and similarities for the ab initio and DFT theoretical descriptions (TD) of the electric (hyper)polarizability of the sodium tetramer. We have a total of nine independent Cartesian components so we have in our hands a 9D problem. The similarities between the theoretical descriptions (TD) of Table 1 are given in Table 2. The quantities shown in Table 2 allow us a direct quantitative evaluation of the similarity of the performance of two arbitrary methods over the calculation of the (hyper)polarizability of the sodium tetramer. In Fig. 4 we show a histogram with the similarity of any method to the presumably most accurate one CCSD(T). As this is our reference value, S(CCSD(T), CCSD(T)) = 1. The method closest to CCSD(T) is CCSD: S(CCSD, CCSD(T)) = 0.86337. Among the DFT methods, the closest to the reference is B3PW91, with S(B3PW91,CCSD(T)) = 0.68609. We are also able to glean more specific information about the relative performance of the various methods. Of all DFT methods, B3PW91 is closest to mPW1PW91, S(B3PW91, mPW1PW91) = 0.90949. PBEPBE is closest to PBEPW91, S(PBEPBE, PBEPW91) = 0.95325 and mPW1PW91 closest to mPW1PBE, S(mPW1PBE, mPW1PW91) = 0.94854.

Table 2 Similarity of the performance of theoretical methods on the electric (hyper)polarizability of Na₄

Fig. 4

Histogram of the similarities S(k,CCSD(T)) where k = ab initio and DFT for the 9D descriptions {α _xx , α _yy , α _zz , γ _xxxx , γ _yyyy , γ _zzzz , γ _xxyy , γ _yyzz , γ _zzxx }

The minimum spanning tree (MST) of the theoretical descriptions of Table 1 is shown in Fig. 5. The tree is a Graph G(p,q) of order 14 and magnitude 13 (p = q + 1). Removing all edges D(i,j) above a threshold of D _T = 0.4, D(i,j) > D _T results in the clustering shown also in Fig. 5. The obtained clusters are as follows:

Fig. 5

Minimum spanning tree (MST) for the theoretical descriptions of the sodium tetramer and subsequent clustering with a distance threshold D _T = 0.4.

• C1 = {SCF}

• C2 = {MP2}

• C3 = {MP3}

• C4 = {DQ-MP4, SDQ-MP4, MP4, CCSD, CCSD(T)}

• C5 = {B3LYP}

• C6 = {PBEPBE, PBEPW91}

• C7 = {B3PW91, mPW1PW91, mPW1PBE}

• $$ {{\rm TD}} = {{\rm C}1} \cup {{\rm C}2} \cup {{\rm C}3} \cup {{\rm C}4} \cup {{\rm C}5} \cup {{\rm C}6} \cup {{\rm C}7} $$

High-level ab initio results form a large cluster C4 = {DQ-MP4, SDQ-MP4, MP4, CCSD, CCSD(T)} which contains the most accurate theoretical descriptions CCSD and CCSD(T). The DFT methods form three distinct clusters C5, C6, and C7. The theoretical description of all DFT closest to CCSD(T) is B3PW91 which belongs to the C7 cluster, B3PW91 ∈ C7.

In Fig. 6 we show the variation of the distance of all DFT methods from the C4 ≡ S cluster that contains all high-level ab initio methods. Instead of comparing DFT methods individually to one ab initio theoretical description we define directly an element-set distance as follows:

Fig. 6

Distance of the DFT theoretical descriptions from the C4 ≡ S cluster that contains the high-level ab initio methods

$$ D(x,S) = \mathop{{\min (x,s)}}\limits_{{s \in S}}. $$

Here, D(x,S) defines the distance of a theoretical description x from the S cluster. In this case S is the cluster that contains all high-level ab initio theoretical descriptions and x a DFT method. In Fig. 6 we see clearly the distance of all DFT methods from the C4 ≡ S cluster. This is a very realistic view of the highly complex matter of the performance of DFT methods. Figure 6 reveals that the DFT method most distant from the S cluster is the B3LYP one. The respective distance is D(B3LYP,S) = 1.242. The methods most close to the S cluster are B3PW91 and mPW1PW91. The respective distances are D(B3PW91,S) = 0.7175 and D(mPW1PW91,S) = 0.6975. Ordering the DFT methods with respect to their distance from S results in the sequence D(B3LYP,S) > D(PBEPW91,S) > D(PBEPBE,S) > D(mPW1PBE,S) > D(B3PW91,S) > D(mPW1PW91,S).

3.2 New Classes of Molecules, the Case of HXeI

HXeI is a typical representative of a fascinating new class of molecules. HRgX compounds are produced by UV radiation of hydrogen halides (HX) in rare gas (RG) matrices [116]. They have attracted considerable experimental and theoretical attention [117]. The importance of the HXeI molecule in particular was brought forth by the work of Buck and Farnik [118]. In their experiments this linear molecule is detected by orientation in strong laser and weak electric fields [119]. Computational experience on HXeI is relatively limited. An empirical estimate of the anisotropy of the dipole polarizability has been proposed by Nahler et al. [120] in their work on the photodissociation of oriented HXeI molecules generated from HI–Xe _n clusters.

In this section we turn our attention on the electric dipole moment, polarizability, and hyperpolarizability of this important species. We lean heavily for molecular data and insights on our recent paper on the electric properties of HXeI [121].

We take into account two classes of molecular properties. Ab initio results calculated with basis sets (given in I/Xe/H order) B5 ≡ [10s9p8d1f/9s8p7d1f/6s3p1d] (179 CGTF) and B9 ≡ [11s10p10d3f/9s8p7d1f/7s5p1d] (214 CGTF). The B9 basis set was used in the DFT calculations. With basis B5 we calculated SCF, MP2, SDQ-MP4, MP4, CCSD, and CCSD(T) and with the larger B9 basis SCF and MP2 values. All ab initio values were taken from the above-cited paper. The DFT results were calculated with basis B9 and the methods B1LYP, B3LYP, B3PW91, mPW1PW91, HCTH, BHandH, BHandHLYP, PBEPBE, and PBEPW91. The molecular geometry of this linear molecule is defined by the bond lengths R(I–Xe) = 3.0577 and R(Xe–I) = 1.7077 Å. It was obtained at the MP2(Full)/B9 level of theory. At the same MP2/B9 level, a natural bond orbital analysis (NBO) yields charges of −0.55803 (I), 0.57691 (Xe), and −0.01888 (H). The calculated independent Cartesian components (z is the molecular axis) of the electric property tensors are given in Table 3.

Table 3 Electric properties of HXeI at the theoretical molecular geometry

The molecular values listed in Table 3 show clearly the difficulty in predicting reliable (hyper)polarizabilities for HXeI. This is particularly evident in the observed variations of the longitudinal components of the first and second hyperpolarizability. We base most of the presentation and analysis in this part on the invariants of (hyper)polarizability: mean values $\bar{\alpha }$, Δα, $\bar{\beta }$, and $\bar{\gamma }$. Figure 7 shows the method dependence of the dipole moment. We also give a few characteristic DFT values. The high-level ab initio data are well grouped together. The DFT method closest to the most accurate CCSD(T)/B5 value is the B3PW91/B9 one which yields μ _z  = 2.4181 ea ₀. The BHandHLYP method predicts a dipole moment close enough to the MP2 one. In Fig. 8 the evolution of the method dependence of the mean dipole polarizability shows a clear gap between SCF and the post-Hartree–Fock methods, a sign of a very strong electron correlation effect. We observe a clear discrepancy between DFT and ab initio methods. Among the most reliable DFT methods the HCTH and B3PW91 yield mean dipole polarizabilities $\bar{\alpha } = 95.87 \ { {\rm and} } \ 93.79{ }{{e}^2}a_0^2E_h^{{ - 1}}$, respectively, both clearly below the most accurate CCSD and CCSD(T) results. The anisotropy of the dipole polarizability is normally a more severe test for DFT methods than that of the mean. In Fig. 9, the method dependence of the anisotropy resembles closely that of the mean. Among all other DFT methods, we note the performance of BHandHLYP, B3PW91, and PBE which give Δα = 103.94, 101.76, and 100.59 e ² a ²₀ E _h ⁻¹, respectively, all three close enough to the SCF value. The ab initio values for the mean first hyperpolarizability are shown in Fig. 10. Of all DFT methods, BHandHLYP performs best, yielding a value $\bar{\beta } = - 949.6{ }{{e}^3}a_0^3E_h^{{ - 2}}$. This result is close to both MP4 and CCSD(T). The performance of other DFT methods is characteristically poor. B3PW91 yields $\bar{\beta } = - 644.8{ }{{e}^3}a_0^3E_h^{{ - 2}}$, a value significantly lower in magnitude than the BHandHLYP one. Last, ab initio and DFT values for the mean second hyperpolarizability are given in Fig. 11. Electron correlation lowers significantly the magnitude of this important property. With the notable exception of BH and HLYP and B3PW91, which predict values close to the SCF one, $\bar{\gamma } = - 50993 \ { {\rm and} } \ 47073{ }{{e}^4}a_0^4E_h^{{ - 3}}$, respectively, all other DFT methods yield considerably higher values.

Fig. 7

Method dependence of the dipole moment of HXeI

Fig. 8

Method dependence of the mean polarizability of HXeI

Fig. 9

Method dependence of the polarizability anisotropy of HXeI

Fig. 10

Method dependence of the mean first hyperpolarizability of HXeI

Fig. 11

Method dependence of the mean second hyperpolarizability of HXeI

We have calculated distance/proximity values for all methods/basis sets used in this section. The respective theoretical descriptions are 8D strings of molecular properties {μ _z , α _zz , α _xx , β _zzz , β _zxx , γ _zzzz , γ _xxxx , γ _xxzz }. The calculated values are given in Table 4. To simplify matters, we have assigned numbers to all methods: 1 = SCF, 2 = MP2, 3 = SDQ-MP4, 4 = MP4, 5 = CCSD, 6 = CCSD(T), 7 = SCF, 8 = MP2, 9 = B1LYP, 10 = B3LYP, 11 = B3PW91, 12 = mPW1PW91, 13 = HCTH, 14 = BHandH, 15 = BHandHLYP, 16 = PBEPBE, and 17 = PBEPW91. To avoid confusion we sometimes denote methods 7 and 8 by 7 = SCF-B9 and 8 = MP2-B9, respectively, to avoid confusion with 1 = SCF and 2 = MP2 results that have been calculated with basis set B5. The most distant or dissimilar theoretical descriptions are 1 = SCF and 6 = CCSD(T): S(1,6) = 0. The two SCF/B5 and SCF/B9 descriptions are very similar, S(1,7) = 0.95157. This is not the case for the MP2/B5 and MP2/B9 descriptions as S(2,8) = 0.55279. In Fig. 12 we show an histogram of the evolution of the similarity S(k,6) ≡ S(k, CCSD(T)). As 6 = CCSD(T) is the reference theoretical description here the maximum similarity for the histogram in Fig. 12 is S(6,6) = 1. The ab initio theoretical description closest to 6 = CCSD(T) is 4 = MP4: S(4,6) = 0.80257. From the DFT descriptions the closest to 6 = CCSD(T) is 9 = B1lYP: S(6,9) = 0.41201. Thus, as one easily gathers from the histogram, the similarity between ab initio (calculated with basis set B5) and DFT descriptions is rather poor. This is also the case for the comparison of ab initio results calculated with basis sets B5 and B9. To view similarity from another perspective, we show in Fig. 13 the evolution of the similarity S(k,11) ≡ S(k, B3PW91). The 11 = B3PW91 method is similar enough to MP2, SDQ-MP4, and CCSD. It is most close to the 12 = mPW1PW91 method: S(11,12) = 0.94834.

Table 4 Similarity of the performance of theoretical methods on the electric (hyper)polarizability of HXeI

Fig. 12

Histogram of the similarities S(k,CCSD(T)) where k = ab initio and DFT for the 8D descriptions {μ _z , α _zz , α _{xx ,} β _zzz, β _zxx , γ _zzzz , γ _xxxx , γ _xxzz }

Fig. 13

Histogram of the similarities S(k,B3PW91) where k = ab initio and DFT for the 8D descriptions {μ _z , α _zz , α _{xx ,} β _zzz, β _zxx , γ _zzzz , γ _xxxx , γ _xxzz }

The minimum spanning tree (MST) corresponding to the graph representing the theoretical descriptions (Table 3) is shown in Fig. 14. This tree is a graph G(p,q) of order p = 17 and size q = p − 1. One expects this graph to be of sufficient complex structure. Clustering by removing all edges greater than a threshold value D _T = 0.4 results in the following partitioning of the spaces of theoretical descriptions:

Fig. 14

Minimum spanning tree (MST) for the theoretical descriptions of HXeI and subsequent clustering with a distance threshold D _T = 0.4

• C1 ≡ {1,7}

• C2 ≡ {2,3,5}

• C3 ≡ {4}

• C4 ≡ {6}

• C5 ≡ {8}

• C6 ≡ {9,10}

• C7 ≡ {11,12}

• C8 ≡ {13,16,17}

• C9 ≡ {14,15}

• $$ {{\rm TD}} = {{\rm C}1} \cup {{\rm C}2} \cup {{\rm C}3} \cup {{\rm C}4} \cup {{\rm C}5} \cup {{\rm C}6} \cup {{\rm C}7} \cup {{\rm C}8} \cup {{\rm C}9} \equiv \bigcup\limits_i {{{C}_i}}, i = 1,2,\ldots, 9 $$

Some characteristics of this clustering are easily discernible. The two SCF and SCF-B9 descriptions form cluster C1. Three high-level ab initio theoretical descriptions form cluster C2 ≡ {2,3,5} ≡ {MP2, SDQ-MP4, CCSD}. The composition of the C6, C7, and C8 clusters grouping DFT methods is clearly understood: B3PW91 is very close to mPW1PW91, BH and H is close to BH and HLYP, and PBEPBE is close to PBEPW91. HCTH belongs to the same cluster as PBEPBE and PBEPW91 and for good reason: it is most similar to these two DFT methods with S(HCTH,PBEPBE) ≈ S(HCTH,PBEPW91) ≈ 0.8.

Last, in Fig. 15 we show the evolution of the distance of the ab initio descriptions calculated with basis set B5 from the reference group of widely used B3LYP, B3PW91, and mPW1PW91. The distance d(x,S), where S = {B3LYP, B3PW91, mPW1PW91} and x ∈ Ω ≡ {SCF, MP2, SDQ-MP4, MP4, CCSD, CCSD(T)}, is shortest for the most accurate CCSD(T) method:

Fig. 15

Distance of the ab initio theoretical descriptions calculated with basis B5 from the reference DFT group S = {B3LYP, B3LYP, mPW1PW91}

$$ d(CCSD(T),S) \le \mathop{{d(x,S)}}\limits_{{x \in \Omega }}. $$

The distance d(x,S) increases as follows:

$$ d({\mathrm{CCSD}}(T),S) < d({\mathrm{SCF}},S) < d({\mathrm{MP}}4,{ ~ }S) < d({\mathrm{CCSD}},S) < d({\mathrm{SDQ}} - {\mathrm{MP}}4,S) < d({\mathrm{MP}}2,{ ~ }S). $$

3.3 Interaction-Induced Polarizability and Hyperpolarizability of Two Water Molecules

In previous work [122] we presented an extended computational study of the interaction-induced electric properties of the water dimer (H₂O)₂. Our findings strongly suggest that the interaction-induced mean dipole polarizability and hyperpolarizability are nearly additive, as

$$ \overline \alpha {{({{{\rm H}}_2}{{\rm O}})}_2} \approx 2\overline \alpha ({{{\rm H}}_2}{{\rm O}}){{\rm and}}\overline \gamma {{({{{\rm H}}_2}{{\rm O}})}_2} \approx 2\overline \gamma ({{{\rm H}}_2}{{\rm O}}). $$

This surprising result appears to hold for the dipole polarizability of certain classes of water clusters, as brought forth in the work of Rodriguez et al. [123] or Ghanty and Ghosh [124]. The above defined additivity results clearly suggests that the respective interaction-induced mean (hyper)polarizability is very small. Identifying the proton acceptor (A) and proton donor (D) in the two moieties of (H₂O)₂ ≡ AD, we define the interaction properties in the counterpoise corrected (CP) scheme as

$$ \begin{gathered} {{\overline \alpha}_{\mathrm{int}}}{{({{\mathrm{H}}_2}{\mathrm{O}})}_2} = \overline \alpha {{({{\mathrm{H}}_2}{\mathrm{O}})}_2} - \overline \alpha ({{\mathrm{H}}_2}{\mathrm{O}} - X) - \overline \alpha (X - {{\mathrm{H}}_2}{\mathrm{O}}), \hfill \\ {{\overline \gamma}_{{{\rm int} }}}{{({{\mathrm{H}}_2}{\mathrm{O}})}_2} = \overline \gamma {{({{\mathrm{H}}_2}{\mathrm{O}})}_2} - \overline \gamma ({{\mathrm{H}}_2}{\mathrm{O}} - X) - \overline \gamma (X - {{\mathrm{H}}_2}{\mathrm{O}}). \hfill \\ \end{gathered} $$

In a very recent paper, we demonstrated that the calculation of the interaction-induced (hyper)polarizability of a molecular system as the water dimer constitutes a sever test of the validity of DFT methods [125]. In this section we add more calculations in order to extend our observations to the performance of a larger class of DFT-based approximations.

Full computational details about the work presented in this section are given in our papers on the water dimer. See Fig. 16 for the relative orientation and respective role of the two water molecules. We adopt a dimer geometry that keeps the monomer H₂O geometry frozen [126]. Thus, comparison of supermolecule to monomer properties makes eminent sense. All calculations presented here have been performed with the water-molecule-specific [6s4p3d1f/4s3p1d] basis set. The ab initio methods employed are SCF, MP2, SDQ-MP4, MP4, CCSD, and CCSD(T). The DFT methods are B1LYP, B3LYP, B3PW91, mPW1PW91, mPW1PBE, PBEPBE, PBEPW91, HCTH, BHandH, and BHandHLYP.

Fig. 16

Relative orientation of the two interacting water monomers in (H₂O)₂ and definition of the interaction-induced properties

In Table 5 we give all monomer and dimer value used in the analysis of the results. The H₂O monomer (M) values pertain to the molecular geometry of the moiety in the dimer. The per-monomer (PM) values are simply defined as PM = AD/2 that is $\overline \alpha {{({{{\rm H}}_2}{{\rm O}})}_2}/2$ for the polarizability and $\overline \gamma {{({{{\rm H}}_2}{{\rm O}})}_2}/2$ for the hyperpolarizability. The properties of interest are the interaction mean (hyper)polarizabilities obtained by the formula AD − AX − XD (see above). A first and valuable observation here is the strong disagreement of ab initio and DFT methods on the (hyper)polarizability of the water dimer. Such behavior for the DFT methods has been noted and analyzed early enough [127]. We must emphasize at this point the essential difference of the BHandH and BHandHLYP DFT methods. Both seem to be very close to the ab initio ones and quite distinct from the other DFT approaches.

Table 5 Analysis of the interaction-induced electric properties for the theoretical equilibrium molecular geometry of the water dimer (AD)

In Fig. 17 we show the histogram of interaction-induced mean polarizability for all methods. With the notable exception of the BHandHLYP method the interaction property ${{\overline \alpha}_{{{\rm int} }}}{{({{{\rm H}}_2}{{\rm O}})}_2}$ is negative for all ab initio methods and positive for the DFT ones. An almost extreme behavior is obvious for the PBEPBE, PBEPW91, and HCTH methods.

Fig. 17

Histogram of the performance of theoretical methods on the calculation of the interaction-induced mean dipole polarizability of the water dimer

The interaction-induced mean hyperpolarizabilities ${{\overline \gamma}_{{{\rm int} }}}{{({{{\rm H}}_2}{{\rm O}})}_2}$ are shown in Fig. 18. The very same pattern as in the case of ${{\overline \alpha}_{{\rm int}}}{{({{{\rm H}}_2}{{\rm O}})}_2}$ is visible here as well. The PBEPBE, PBEPW91, and HCTH methods yield very large values for the interaction hyperpolarizability. The BHandH and BHandHLYP methods again give surprisingly small ${{\overline \gamma}_{{{\rm int} }}}{{({{{\rm H}}_2}{{\rm O}})}_2}$ values.

Fig. 18

Histogram of the performance of theoretical methods on the calculation of the interaction-induced mean second hyperpolarizability of the water dimer

Last, in Fig. 19 we show the evolution of the DFT values for the interaction-induced hyperpolarizability for the monomer (M) and the per-monomer (PM) value. In fact the ${{\rm PM}} = \overline \gamma {{({{{\rm H}}_2}{{\rm O}})}_2}/2{{\rm and}}\;M = \overline \gamma ({{{\rm H}}_2}{{\rm O}})$ values determine the differential-per-monomer hyperpolarizability, defined as

Fig. 19

Mean second hyperpolarizability for the water monomer (M) and dimer (per monomer PM) for all DFT methods. Reference CCSD(T) values are also included

$$ {{\rm DHPM}} = [\overline \gamma {{({{{\rm H}}_2}{{\rm O}})}_2} - 2\overline \gamma ({{{\rm H}}_2}{{\rm O}})]/2 = \overline \gamma {{({{{\rm H}}_2}{{\rm O}})}_2}/2 - \overline \gamma ({{{\rm H}}_2}{{\rm O}}) \equiv {{\rm PM}} - M. $$

In Fig. 19 we also show the reference M and P values calculated at the CCSD(T)/[6s4p3d1f/4s3p1d] level of theory. We observe that the sequence B3LYP, B3PW91, mPW1PW91, mPW1PBE displays a smooth convergence towards the reference CCSD(T) values. The PBEPBE, PBEPW91, and HCTH methods are characterized by very large M and PM values. Obviously, the BHandH and BHandHLYP methods appear as something of an anomaly here. Both the PM and M values for these two methods are lower than the respective reference CCSD(T) values.

4 Final Remarks and Conclusions

We have investigated and closely analyzed the predictive capability of DFT methods in the calculation of electric polarizabilities and hyperpolarizabilities for three difficult cases: the sodium tetramer, the new compound HXeI, and the interaction (hyper)polarizability of the two moieties in the weakly bonded water dimer. In the first two cases we have shown that the ab initio and DFT methods offer a quite different picture of the electric polarizability and hyperpolarizability. In the case of (H₂O)₂ we are led to two drastically divergent pictures as the two classes of methods clearly differ even in the sign of the interaction-induced mean polarizability and hyperpolarizability.

We show that it is probably a distinct advantage to talk about the quality or predictive capability of a set of DFT methods instead of trying to obtain a picture of the potential of a single one. This strongly corroborates our recent strategy to rely on a given set of DFT methods in order to form a clear idea about the DFT perspective in electric property calculations [115, 128, 129].