Abstract
 Top of page
 Abstract
 Introduction
 The NESC Method
 NESC Analytic Energy Derivatives
 Second Analytic Derivatives and SecondOrder Response Properties
 Conclusions and Outlook
The normalized elimination of the small component method is a first principles twocomponent relativistic approach that leads to the Diracexact description of oneelectron systems. Therefore, it is an ideal starting point for developing procedures, by which first and secondorder response properties can be routinely calculated. We present algorithms and methods for the calculation of molecular response properties such as geometries, dipole moments, hyperfine structure constants, vibrational frequencies and force constants, electric polarizabilities, infrared intensities and so forth. The described formalisms are applied to molecules containing mercury and other heavy elements, which require a relativistic treatment. Perspectives for the future development and application of Diracexact methods are outlined. © 2013 Wiley Periodicals, Inc.
Introduction
 Top of page
 Abstract
 Introduction
 The NESC Method
 NESC Analytic Energy Derivatives
 Second Analytic Derivatives and SecondOrder Response Properties
 Conclusions and Outlook
The influence of relativistic effects[1] on the electronic structure and the chemical properties of heavy elements was recognized more than four decades ago.[213] At the quantum mechanical level, the Dirac equation[14, 15] provides the basis for an exact description of a single electron in an external field, and the Dirac Hamiltonian is a cornerstone of relativistic quantum chemistry. The Dirac equation considers the electron spin as a dynamic variable and treats the chargeconjugate particle (positron) on the same footing as the electron itself. This leads to a fourcomponent wavefunction, which, in the standard representation, comprises the largecomponent and the smallcomponent spinors, and imposes a special requirement on the basis sets used in relativistic quantum chemical schemes.[16, 17] To guarantee the correct kinetic energy of electrons, the basis functions and for the largecomponent and the smallcomponent spinors, respectively, should satisfy the kinetic balance condition[16] given by Eq. (1)
 (1)
in which is the vector of Pauli matrices and is the linear momentum operator. The necessity to impose the kinetic balance condition leads to a rapid increase of the number of basis functions and respective molecular integrals required in the relativistic quantum chemical calculation and makes these calculations considerably more time consuming than the corresponding nonrelativistic ones.[7]
The complexity of the fourcomponent relativistic formalism can be bypassed by switching to the twocomponent (quasi) relativistic approach, which is achieved by decoupling the electronic and positronic states and by keeping the explicit description for electrons only.[18] However, with the exception of a free electron, the exact algebraic form of such a transformation, the socalled Foldy–Wouthuysen (FW) transformation,[18] is not known. Several approaches were developed to tackle the problem of transforming the fourcomponent relativistic formalism to a twocomponent one.[19, 20] From the very beginning in the midseventies, the field of (quasi) relativistic twocomponent computational schemes was dominated by an operatordriven approach where the (exact or approximate) twocomponent relativistic Hamiltonian is first expressed in operator form and then transformed to matrix form suitable for quantum chemical calculations. A widely used formalism developed by Douglas and Kroll[19] and later extended by Hess (DKHmethod)[2124] uses a factorization of the unknown FW transformation operator into a freeparticle part, which is known exactly, and a fielddependent part, which is approximated to a certain order in the interaction strength. This leads to a convergent series of approximations which, however, uses a large number of auxiliary equations for the intermediate transformation operators and, when converted to matrix form, results in a huge number of matrix operations.[25, 26] Therefore, it is not surprising that DKH analytic energy derivatives have been developed only recently and then, only for the loworder approximation of order 2 (DKH2).[27, 28]
An alternative to the operatordriven formulation of the twocomponent relativistic methodology was proposed by Dyall[29] who was probably the first to carry out all the pertinent transformations in matrix form starting from the matrix Dirac equation.[2932] The resulting normalized elimination of the small component (NESC) formalism is computationally simple and transparent and enables one to obtain the exact electronic (positiveenergy) solutions of the Dirac equation.[29, 33, 34] The initial success of the NESC methodology has led to a subsequent development of alternative matrixdriven twocomponent methods,[3538] which have been shown to be equivalent to NESC.[39] It is also noteworthy that approximate relativistic methods based on the socalled regular approximation[20, 4043] can be easily derived from NESC, which leads to a simple matrix formulation of these methods.[44, 45]
An important advantage of the NESC methodology is the availability of analytic energy derivatives,[4649] which enable one to obtain relativistically corrected atomic and molecular properties via the response formalism.[5054] The primary purpose of this article is to provide a succinct overview of the NESC analytic derivatives formalism and its application to the calculation of various relativistically corrected molecular properties. We start by recapitulating the salient features of the NESC methodology and continue by introducing the firstorder NESC energy derivatives formalism and its application to the calculation of properties such as the analytic gradient for the accurate and routine optimization of molecular geometries, the determination of contact electron densities and electric field gradients (EFGs) at nuclear positions, or the calculation of magnetic hyperfine structure (HFS) constants. In the subsequent section, the secondorder NESC energy derivatives formalism and its application to the calculation of vibrational frequencies, infrared (IR) intensities, and molecular polarizabilities will be described. In the last section, conclusions will be drawn and some thoughts about the future prospects of the NESC methodology will be outlined.
The NESC Method
 Top of page
 Abstract
 Introduction
 The NESC Method
 NESC Analytic Energy Derivatives
 Second Analytic Derivatives and SecondOrder Response Properties
 Conclusions and Outlook
The derivation of the NESC equations starts from the oneelectron Dirac equation represented in matrix form and modified in such a way that the socalled restricted kinetic balance condition is folded into the equation.[2932] The latter is achieved by introducing[29] a pseudolarge component of the relativistic wavefunction via Eq. (2),
 (2)
which connects it to the small component used in the traditional formulation of the Dirac equation. Although not unique, the definition of the pseudolarge component enables one to eliminate the dependence of the relativistic metric on the spin and to partition the modified Dirac equation into the spinfree and spindependent (spinorbit) parts in a simple and transparent way as was first discussed by Kutzelnigg[31, 32] and later used by Dyall.[30] In matrix form, the modified Dirac equation is given in Eq. (3),
 (3)
where A and B are the matrices collecting the expansion coefficients of the large and pseudolarge components of the relativistic wavefuction in terms of the basis functions (or for brevity) and the diagonal matrix contains the energy eigenvalues. The superscripts or subscripts and denote the positive (electronic) and negative (positronic) eigenvalue and eigenvector solutions of the equation. Symbols S, T, and V represent the nonrelativistic overlap, kinetic energy, and potential energy matrices, respectively, and W is the matrix of the operator . With the use of the Dirac identity , the spinfree and spinorbit parts of the W matrix can be separated as in Eq. (4),
 (4)
which leads to the commonly used spinscalar approximation resulting from the neglect of the spindependent part . In the following, the spinscalar approximation will be used (unless noted otherwise) and superscript sf of matrix will be dropped in cases without ambiguity.
By introducing a matrix U that connects the large and the pseudolarge components via Eq. (5),
 (5)
In Eqs. (7b) and (7c), is the ESC matrix in the infiniteorder regular approximation.[45] The iterative solution can be achieved by using a damped fixed point iteration technique,[34] which, if started from an appropriate guess, requires fewer number of floating point operations than the onestep method. The latter method, however, offers a better stability, especially in cases where a large number of very tight basis functions are used.
The eigenfunctions and eigenvalues of the NESC method are fully equivalent to the electronic solutions of the oneelectron Dirac equation.[29, 34] For manyelectron systems, it is possible to derive, when starting from the modified Dirac–Coulomb equation,[55, 56] a similar set of equations to obtain electronic states only.[29] However, this leads to the necessity of calculating a large number of twoelectron integrals in addition to those required by a nonrelativistic Hartree–Fock (HF) calculation. A simpler alternative is to consider relativistic effects in the oneelectron part of the manyelectron Hamiltonian only and treat the electron–electron interactions nonrelativistically.[57] As has been shown by Dyall,[57] the oneelectron approximation defined in this way is a sufficiently accurate approximation to the manyelectron relativistic selfconsistent field (SCF) approach in the FW representation. Within the oneelectron ( ) approximation, the atomic or molecular Fock operator is given by Eq. (9),
 (9)
where J and K correspond to the Coulomb and exchange contributions to the twoelectron part of the nonrelativistic Fock operator. The total energy of a manyelectron system in the approximation is then given by Eq. (10),
 (10)
where is the density matrix constructed using the eigenvectors C of the Fock operator (9) and the diagonal matrix of the orbital occupation numbers n.
The renormalization matrix G takes care of the transformation of the oneelectron NESC Hamiltonian from the relativistic normalization of the electronic wavefunction to the nonrelativistic normalization. The G matrix possesses correct transformation properties under linear transformations of the basis set. It is calculated as the square root of as given in Eq. (11)
 (11)
which was derived by Peng and Liu.[58] The NESC formalism described can be easily implemented in the existing nonrelativistic quantum chemical codes as it does not require the calculation of new molecular integrals over basis functions. Investigations based on NESC require essentially the same elapsed central processing unit (CPU) time as the corresponding nonrelativistic calculations. The NESC method provides the exact quantum mechanical description of any oneelectron system and by this it is fully equivalent to the Dirac fourcomponent method. Hence, NESC is termed a Diracexact method. Further details on the implementation of the NESC method can be found in the original publication.[34]
NESC Analytic Energy Derivatives
 Top of page
 Abstract
 Introduction
 The NESC Method
 NESC Analytic Energy Derivatives
 Second Analytic Derivatives and SecondOrder Response Properties
 Conclusions and Outlook
When differentiating the NESC total energy (10) with respect to an arbitrary external perturbation parameter λ, one obtains Eq. (12)
 (12)
where is the energyweighted density matrix (Lagrangian) and the prime at implies that only the twoelectron integrals have to be differentiated.[46]
Equations (13) and (14) depend on the molecular integral derivatives already available in most of the nonrelativistic quantum chemical codes. The only new terms for which derivatives need to be developed are the last two terms in Eq. (13b) and the terms in Eq. (14b), which depend on the derivatives of the renormalization matrix and of the ESC matrix, respectively.[46]
Using Eq. (15), the derivatives of the square root of a matrix are calculated element by element and stored to external memory for later use in the gradient calculation.[27, 48] Such an approach is computationally inefficient and may represent a bottleneck in the gradient computation. A more efficient computational strategy[46] is based on the fact that the derivatives of contribute to the final gradient in the form of traces of matrix products [see Eq. (13b)]. Exploiting the cyclic property of the trace, that is, , these contributions can be transformed according to Eq. (16),
 (16)
where two new matrices are introduced: Z with elements and . With the use of Eq. (12), the contributions of the renormalization matrix derivatives to the final gradient, that is, the last two terms in Eq. (13b), are given by Eq. (17)
 (17a)
 (17b)
where the matrices , , and do not depend on the perturbation λ and their calculation requires only a few matrix multiplications (see the original publication[46] for further detail). Thus, the algorithm for obtaining the contributions of the derivatives of the renormalization matrix is formulated entirely in terms of traces of matrix products, which makes it convenient for the implementation in existing nonrelativistic quantum chemical codes.[46]
The NESC analytic derivatives formalism[46, 50] requires only a modest computational effort and can be easily implemented in the nonrelativistic quantum chemical codes. The formalism was tested by comparing analytically and numerically obtained energy gradients for a number of molecules containing heavy elements. As demonstrated in Table 1, the formalism developed leads to exact energy derivatives and requires only a fraction of the CPU time elapsed for a single SCF iteration. In the following subsections, the NESC analytic derivatives formalism is applied to obtain molecular geometries and other firstorder response properties.
Geometry optimizations using the NESC analytic gradient
The NESC analytic energy gradient was applied to determine molecular geometries of molecules containing heavy atoms such as Hg, Tl, I, or Au.[46, 60] Zou, Filatov, and Cremer (ZFC)[46] used the analytic NESC energy gradient in connection with the NESC/CCSD (coupled cluster with single and double substitutions) method for geometry optimizations and the NESC/CCSD(T) (CCSD with perturbational treatment of triple substitutions) to obtain bond dissociation energies (BDE) at the NESC/CCSD geometries. A selection of the calculated geometries and BDEs is given in Table 2.
Table 2. NESC/CCSD geometries and NESC/CCSD(T) bond dissociation energies D_{e} (enthalpies D_{0}) of mercury molecules.aMolecule  Sym  State  Method  Geometry parameters  D_{e} (D_{0})  Reference 


HgF   ^{2}  NESC/CCSD(T)//NESC/CCSD  2.024  33.0 (32.3)  [46] 
   Expt.   32.9  [61] 
HgCl   ^{2}  NESC/CCSD(T)//NESC/CCSD  2.402  23.8 (23.4)  [46] 
   SOC/ECP/CCSD(T)  2.354  22.9  [62] 
   Expt.  2.395, 2.42  23.4, 24.6  [6365] 
HgBr   ^{2}  NESC/CCSD(T)//NESC/CCSD  2.546  20.0 (17.5)  [46] 
   SOC/ECP/CCSD(T)  2.498  16.3  [62] 
   Expt.  2.62  17.2, 18.4  [6668] 
HgI   ^{2}  NESC/CCSD(T)//NESC/CCSD  2.709  12.9 (7.6)  [46] 
   SOC/ECP/CCSD(T)  2.708  8.6  [62] 
   Expt.  2.81  7.8, 8.1, 8.9  [64, 65, 69] 
HgCN   ^{2}  NESC/CCSD(T)//NESC/CCSD  HgC: 2.118, CN: 1.161  36.1  [46] 
   IORA/QCISD  HgC: 2.114, CN: 1.179   [70] 
HgNC   ^{2}  NESC/CCSD(T)//NESC/CCSD  HgN: 2.077, NC: 1.176  22.4  [46] 
HgCH_{3}   ^{2}A_{1}  NESC/CCSD(T)//NESC/CCSD  HgC: 2.344, HC: 1.084, HgCH: 104.3  3.2  [46] 
It was found[46] that the NESC/CCSD geometries are close to the experimental geometries (see HgCl and HgBr in Table 2) with deviations of the order of 0.1 Å or less. The agreement between NESC/CCSD(T) and experimental bond dissociation enthalpies D_{0} for mercury halides was excellent in view of a mean deviation of just 0.3 kcal/mol.[46]
A number of studies have been published since then, which confirm the reliability of molecular geometries optimized using NESC in connection with either density functional theory (DFT) or coupled cluster theory.[47, 5053, 60, 71]
NESC contact density and Mösbauer isomer shift
The contact density, which in nonrelativistic quantum theory is defined as the electron density at the nuclear position, is used for the interpretation[7274] of the shift of the resonance absorption line in nuclear γresonance spectroscopies, such as the Mössbauer spectroscopy[75] or synchrotron nuclear forward scattering.[76] During the nuclear γtransition, the charge radius of the nucleus changes and this leads to a slight variation of the electronnuclear interaction that can be sensed by experimental measurements.[73, 74] The key electronic structure parameter that defines the magnitude of the resonance line shift (the socalled isomer shift) is the contact density that, at a fully relativistic level of description, can be calculated as a derivative of the total electronic energy with respect to the nuclear charge radius.[77] According to linear response approach proposed by Filatov,[77, 78] the isomer shift δ (and contact density , see below) is given by Eq. (21)
 (21)
where E^{a} and E^{s} denote the electronic energy of the absorbing and source systems, respectively.
A fully analytic approach to obtaining effective contact densities within the linear response formalism based on the NESC method was presented by Filatov, Zou, and Cremer (FZC).[50] For this purpose, the point nucleus (pn) model was extended to a finite nucleus (fn) model. Based on the assumption of a Gaussian nuclear charge distribution in Eq. (22a),
 (22a)
 (22b)
 (22c)
the nucleuselectron attraction potential V adopts the form of Eq. (22b) where the exponential parameter ζ related to the nuclear charge radius R_{K} of Kth nucleus is given by Eq. (22c). Then, the effective contact density is given by[77, 78]
 (23)
in which is the value of the parameter obtained from the experimentally measured rootmeansquare charge radius of the resonating nucleus a. The effective contact density a can be directly compared to the contact density calculated within a traditional approach as the expectation value of the electron density operator at the nuclear position.[50, 77, 78] In the context of the NESC method, the energy derivative in Eq. (23) is given by Eq. (24),
 (24)
where matrices , , and are defined in Eqs. (13) and (20).[46, 50]
Using the equations given above, ZFC[50] investigated the contact densities (in e/bohr^{−3}) of the Hg nucleus in free mercury and in a series of mercury compounds, where in the latter case the contact density differences ρ^{–}_{Hg}−ρ^{–}_{mol} were determined (see Table 3).[50] Trends in the calculated contact density differences were reasonably reproduced already at the NESC/HF and NESC/MP2 (secondorder Møller–Plesset perturbation) levels of theory although the NESC/CCSD represented the most reliable values.[50]
Table 3. Effective contact densities (e/bohr^{−3}) of the Hg atom obtained at the NESC level of theory.Atom/molecule  NESC/HF  NESC/MP2  NESC/CCSD 


Hg  2104944.971  2105047.821  2105035.382 
Hg^{+}  112.876  127.943  121.136 
Hg^{2+}  278.394  305.695  293.217 
HgF  98.086  81.294  76.872 
HgF_{2}  121.352  108.368  104.387 
HgF_{4}  96.586  109.453  96.264 
HgCl_{2}  108.118  94.572  91.592 
Hg(CH_{3})_{2}  49.001  43.610  42.184 
 240.820  245.550  237.066 
The values of the contact density differences are large when the electronic environment strongly differs from that of the free Hg atom. The differences stretch from about 40 e/bohr^{−3} in dimethyl mercury to 293 e/bohr^{−3} in the mercury dication. With increasing electronegativity of the Hgsubstituents, the difference contact density increases to 104 e/bohr^{−3}, which confirms that the contact densities and Mössbauer isomer shifts δ are sensitive probes of the electronic environment and coordination sphere of the Hg nucleus.[50]
Nuclear quadrupole interaction and EFG
Using the NESC analytic derivatives formalism, FZC[51] expressed the expectation values of the components of the EFG tensor as derivatives of the NESC total energy with respect to the quadrupole tensor components of the Kth nucleus:
 (29)
Using the NESC EFG formalism in connection with correlation corrected ab initio methods, FZC[51] investigated a series of molecules and found that the EFG values obtained are in good agreement with measured values as well as the results as the fourcomponent relativistic calculations, especially. Table 4 compares the EFG principal values obtained at the NESC/HF and the NESC/MP2 level of theory with fourcomponent Dirac–Coulomb HF and CCSDT results.[88] The NESC/MP2 EFG values are in an excellent agreement with the state of the art 4cDCCCSDT calculations.[88] Using the ^{199}Hg NQM value of 0.675 ± 0.012 barn,[88] the NESC/MP2 calculation yields 2414 ± 43 MHz for the NQCC of Hg(CH_{3})_{2}, which is in good agreement with the NQCC of 2400 MHz obtained by PAC in frozen neat Hg(CH_{3})_{2}.[88] The convincing performance of NESC/MP2 is partly a result of the electron correlation corrections, which contribute up to 40% to the final EFG value (see Table 4).
Table 4. NESC electric field gradients (a.u.) on Hg nucleus calculated at the HF and MP2 levels in comparison with fourcomponent Dirac–Coulomb values from Ref. [88].Molecule  4cDCHFa  NESC(pn)/HFb  NESC(fn)/HFc  4cDCCCSDTd  NESC(fn)/MP2 


HgCl_{2}  −12.95  −12.14  −12.12  −9.51  −9.32 
HgBr_{2}  −11.82  −11.11  −11.09  −8.63  −8.54 
HgI_{2}  −11.68  −11.04  −11.03  −8.61  −8.64 
Hg(CH_{3})_{2}  −19.83  −18.77  −18.78  −15.71  −15.22 
Although the NESC/HF data in Table 4 does not show a pronounced dependence on the nuclear charge distribution (the pn and fn results are nearly the same), the formalism developed offers the possibility of investigating the dependence of the EFG tensor on the nuclear size. It has been suggested by Pyykkö[89] that, in a series of compounds of two different isotopes I_{1} and I_{2} of the same element Z, the ratio of the NQCCs may vary from compound to compound, thus revealing a nuclear quadrupole anomaly given by Eq. (31).
 (31)
Magnetic hyperfine structure constants
The NESC HFS formalism is capable of predicting the values, which are in good agreement with the experiment and the results of highlevel fourcomponent relativistic calculations as is reflected by the data in Table 5. In the calculations reported by Hauser et al.,[94] a finite nuclear charge distribution and finite distribution of the nuclear magnetic moment were used in connection with large uncontracted basis sets of pentuplezeta quality. Although the magnetic and charge nuclear radii are generally different (see e.g., Ref. [95]), the former are not available for most elements and accordingly, one uses the same radius for both electric charge and magnetic dipole distributions.[93]
Table 5. Comparison of measured isotropic HFS constants (in MHz) of alkali metals with NESC/CCSD or fourcomponent Dirac–Coulomb CISDpT values[93] with experimental dataAtom  Exp.a  NESC/CCSDb  4cDCCISDpTc 


^{7}Li  401.7  402.0  – 
^{23}Na  885.8  880.2  888.3 
^{39}K  230.8  232.1  228.6 
^{85}Rb  1011.9  1019.1  1011.1 
The accuracy of the NESC HFS calculations makes it possible to estimate the nuclear magnetic radii from a comparison of calculated and measured HFS constants. Table 6 compares values obtained by NESC[52] and fourcomponent DFT calculations[96] with the corresponding experimental data. The NESC values of Table 6 indicate the importance of electron correlation for obtaining reliable HFS constants. Using a finite nuclear model leads to more accurate NESC values. Relativistic DFT calculations noticeably underestimate the magnitude of and strongly depend on the nuclear model used. The hyper sensitivity of the DFT calculations to the nuclear model used is due to an incorrect behavior of the exchangecorrelation potential in the vicinity of the nucleus.[101] The ab inito NESC calculations are free of such drawbacks and yield accurate values, which can be used for refining models of the distribution of the nuclear magnetic moment. To explore this possibility, it is necessary to extend the existing spinscalar NESC analytic derivatives formalism by including spinorbit coupling (SOC) and using a twocomponent relativistic approach.
Table 6. Comparison of calculated isotropic HFS constants (in MHz) of mercury compounds with experimental dataMolecule  Exp.  Nuc. modela  NESC/CCSD  NESC/MP2  NESC/HF  4cDKS/BP86b 


HgH  6859c; 7198d  pn  7463  6616  8238  6921 
fn  7332  6500  8093  6244 
HgF  22163e  pn  20558  21790  23188  18927 
fn  20198  21408  22782  16895 
HgCN  15960f  pn  16135  19766  17341  15599 
fn  15853  19420  17037  13967 
HgAg  2723g  pn  2962  2873  2713  3690 
fn  2910  2822  2665  3285 
Second Analytic Derivatives and SecondOrder Response Properties
 Top of page
 Abstract
 Introduction
 The NESC Method
 NESC Analytic Energy Derivatives
 Second Analytic Derivatives and SecondOrder Response Properties
 Conclusions and Outlook
IR spectra: Vibrational frequencies and intensities
A reliable prediction of IR spectra is needed in connection with the identification and structure description of unknown compounds. This task becomes especially challenging for compounds containing heavy elements for which the effect of relativity on the vibrational frequencies and IR intensities must be included. Often, accurate quantum chemical calculations represent a sole source of absolute IR intensities for large molecules as, most commonly, only relative intensities are measured[104] and experimental data on the absolute intensities are scarce.
Using NESC in connection with DFT calculations utilizing the PBE0 hybrid density functional,[106] ZFC[53] calculated IR spectra for a series of compounds of heavy elements. Table 7 compares the calculated geometries, vibrational frequencies, and IR intensities with the available experimental data, some of which have been measured in solid state samples. The vibrational frequencies calculated are in good agreement with the experimental data considering that the calculated harmonic frequencies were not scaled to approximately include anharmonicity effects. In the case of UF_{6}, measured absolute IR intensities are available.[112] The symmetrical vibrational modes at 184 (exp.: 186) and 629 (exp.: 624) cm^{−1} with intensities of 52 (exp.: 38) and 810 (exp.: 750) km/mol are in a good agreement with the experimental values (see Table 7). It is noteworthy that apart from being essential for identifying unknown compounds via their IR spectra, the IR intensities are also useful for deriving effective atomic charges.[116]
Table 7. Comparison of NESC/PBE0 geometries (distances in Å), harmonic vibrational frequencies (cm^{−1}), and IR intensities (km/mol) with the corresponding experimental values measured in the gas or the solid phase (the latter are indicated by the word solid).aMol. (sym.)  Method  Geometry  Frequency (infrared intensity, mode symmetry) 


AuH ( )  NESC/PBE0  AuH: 1.530  2283.7 (14.7; σ^{+}) 
Expt.[61]  AuH: 1.524  2305.0 (σ^{+}) 
( )  NESC/PBE0  AuH: 1.652  773.8 (115.7; ), 1685.2 (1035.8; ), 1994.9 (0; ) 
Expt.[107]   1636.0 ( ) 
( )  NESC/PBE0  AuH: 1.631  776.4 (0; ), 793.9 (66.6; e_{u}), 828.7 (42.3; ), 843.1 (0; ) 
 1780.6 (2318.0; e_{u}), 2113.7 (0; ), 2118.1 (0; ) 
Expt.[107]   1676.4 (e_{u}) 
AuF ( )  NESC/PBE0  AuF: 1.923  556.7 (52.3; ) 
Expt.[108]  AuF: 1.918  563.7 ( ) 
( )  NESC/PBE0  AuF: 1.963  184.4 (25.0; ), 516.3 (0; ), 548.1 (182.7; ) 
( )  NESC/PBE0  AuF: 1.916  184.0 (0; ), 217.7 (0; ), 233.1 (25.8; ), 253.8 (8.7; e_{u}) 
572.0 (0; ), 597.1 (0; ), 613.4 (383.9; e_{u}) 
Expt. (solid)[109]   230 ( ), 561 ( ), 588 ( ) 
ThO ( )  NESC/PBE0  ThO: 1.826  926.1 (245.7; ) 
Expt.[110]  ThO: 1.840  895.8 ( ) 
Th_{2}O_{2} ( )  NESC/PBE0  ThO: 2.089  155.7 (4.8; ), 192.7 (0; a_{g}), 373.1 (0; ), 527.2 (35.2; ), 
OThO: 74.4  623.8 (297.9; ), 633.9 (0; a_{g}) 
Expt.[111]   619.7 ( ) 
UF_{6} (O_{h})  NESC/PBE0  UF: 1.994  139.1 (0; ), 184.0 (51.7; ), 199.3 (0; ), 539.1 (0; e_{g}) 
 629.4 (810.3; ), 681.4 (0; ) 
Expt.[112, 113]  UF: 1.996  142 ( ), 186.2 (~38; ), 202 ( ), 532.5 (e_{g}), 624 (750; ) 
 667.1 ( ) 
OsO_{4} (T_{d})  NESC/PBE0  OsO: 1.686  352.6 (22.4; t_{2}), 356.5 (0; e), 1031.9 (465.8; t_{2}), 1063.9 (0; a_{1}) 
Expt.[114, 115]  OsO: 1.711  322.7 (t_{2}), 333.1 (e), 960.1 (t_{2}), 965.2 (a_{1}) 
^{265}HsO_{4} (T_{d})  NESC/PBE0  HsO: 1.757  316.1 (32.9; t_{2}), 335.2 (0; e), 1010.4 (463.2; t_{2}), 1056.2 (0; a_{1}) 
Static electric dipole polarizabilities
Another secondorder response property important for the characterization of the electronic structure of molecules and intermolecular interactions is the static electric dipole polarizability, henceforth just called polarizability. The polarizability provides a measure of the distortion of the electric charge distribution in an atom or a molecule exposed to an external electric field.[117] Knowledge of atomic and molecular polarizabilities is important in many areas of chemistry ranging from electron and vibrational spectroscopy to molecular modeling, drug design, and nanotechnology. Especially, for molecules containing relativistic atoms, measured values of polarizabilities are sparse and, therefore, their reliable prediction with the help of relativistic quantum chemical methods is desirable.
As a result of rotational averaging, the scalar isotropic polarizability is typically obtained by measurements is the gas phase.
The polarizability of osmium tetroxide calculated with the NESC/MP2 method (8.23 Å^{−3}) deviates only slightly from the experimental value of 8.17 Å^{−3}.[118] The NESC/MP2 value is in better agreement with experiment than the NESC/PBE0 polarizability of 6.89 Å^{−3},[53] which is too small by more than 1 Å^{−3}. In general, the NESC/MP2 method is more reliable when calculating polarizability values, whereas DFT polarizabilities can be only used when discussing general trends. NESC/MP2 polarizability calculations can be used to correct unreliable experimental values. For example, the measured isotropic polarizability of UF_{6} was reported to be 12.5 Å^{3},[119] which is far too large in view of a NESC/MP2 value of 8.03 Å^{3} (Table 8). The value of (HgCl_{2}) was given in the literature as 11.6 Å^{3},[119] whereas the calculated NESC/MP2 value is 8.60 Å^{3}.[53]
Conclusions and Outlook
 Top of page
 Abstract
 Introduction
 The NESC Method
 NESC Analytic Energy Derivatives
 Second Analytic Derivatives and SecondOrder Response Properties
 Conclusions and Outlook
The derivation of the NESC equations published by Dyall almost two decades ago[29] delineated a new direction of development in the domain of relativistic quantum chemistry. For the first time, the matrixdriven approach to the development of exact and approximate twocomponent relativistic theories was formulated in a concise way and this demarcated a paradigm shift away from operatordriven approaches, which were dominating the field at that time. It took about a decade to realize the full extent of advantages offered by the matrixdriven approach and the NESC method of Dyall triggered article by other researchers[3538, 58, 120, 121] in an attempt to extend or reformulate the method. Currently, the development of matrixdriven quasirelativistic computational methods is an active field of research that holds a considerable promise for the computational modeling of molecules and chemical reactions involving heavy and superheavy elements.
In this review, we have surveyed the most recent developments in the framework of the NESC methodology, which was extended by adding extra functionality in form of analytic energy derivatives,[46, 47] the availability of which lays down the basis for calculating first and secondorder response properties of molecules containing heavy atoms.[5053] The major advantage of the formalism presented is in its computational efficiency, which enables one to calculate relativistically corrected properties of large and very large molecular systems. Although the analytic derivatives formalism presented was formulated within the scalarrelativistic approximation, recent article by FZC[122] lays the basis for its extension to a genuine twocomponent form and the calculation of SOC effects on molecular properties.
Apart from the extension of the response property formalism by taking SOC effects into account, the development of higher order derivatives and higher order response properties can be foreseen for the future. Especially, the extension of the existing NESC first and secondanalytic derivatives formalism to the calculation of magnetic properties measured NMR or ESR spectroscopy such as the magnetic shielding tensor, nuclear spin–spin coupling constants, or the electronic gtensor are within reach. A routine calculation of Raman activities and curvature coupling coefficients (derivatives of the normal vibrational modes) will become feasible for the compounds of heavy elements after the development and implementation of a NESC thirdorder analytic derivatives formalism. Already in its current form, the NESC method can serve as a solid basis for the computational investigation of the properties of molecules containing relativistic elements. Any future extension of the algorithms currently available will strengthen the position of NESC as a generally applicable Diracexact relativistic method.