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A fully automated parallelized implementation of the incremental scheme for coupled-cluster singles-and-doubles (CCSD) energies has been extended to treat molecular (unrelaxed) first-order one-electron properties such as the electric dipole and quadrupole moments. The convergence and accuracy of the incremental approach for the dipole and quadrupole moments have been studied for a variety of chemically interesting systems. It is found that the electric dipole moment can be obtained to within 5% and 0.5% accuracy with respect to the exact CCSD value at the third and fourth orders of the expansion, respectively. Furthermore, we find that the incremental expansion of the quadrupole moment converges to the exact result with increasing order of the expansion: the convergence of nonaromatic compounds is fast with errors less than 16 mau and less than 1 mau at third and fourth orders, respectively (1 mau=10(-3)ea(0)(2)); the aromatic compounds converge slowly with maximum absolute deviations of 174 and 72 mau at third and fourth orders, respectively.

An accurate set of benchmark rotational g tensors and magnetizabilities are calculated using coupled-cluster singles-doubles (CCSD) theory and coupled-cluster single-doubles-perturbative-triples [CCSD(T)] theory, in a variety of basis sets consisting of (rotational) London atomic orbitals. The accuracy of the results obtained is established for the rotational g tensors by careful comparison with experimental data, taking into account zero-point vibrational corrections. After an analysis of the basis sets employed, extrapolation techniques are used to provide estimates of the basis-set-limit quantities, thereby establishing an accurate benchmark data set. The utility of the data set is demonstrated by examining a wide variety of density functionals for the calculation of these properties. None of the density-functional methods are competitive with the CCSD or CCSD(T) methods. The need for a careful consideration of vibrational effects is clearly illustrated. Finally, the pure coupled-cluster results are compared with the results of density-functional calculations constrained to give the same electronic density. The importance of current dependence in exchange–correlation functionals is discussed in light of this comparison.

A general method is presented for the efficient elimination of response parameters in molecular property calculations for variational and nonvariational energies. For variational energies, Wigner's 2n+1 rule is obtained as a special case of the more general k(2n+1) rule, which states that for a subset of k perturbations within a total set of zor=k perturbations, response parameters may be eliminated according to the 2n+1 rule (normally applied to the full set of perturbations). Nonvariational energies may be treated by introducing Lagrange multipliers that satisfy the stronger 2n+2 rule for the k perturbations, while the wave-function parameters still satisfy the 2n+1 rule for the k perturbations. The corresponding rule for nonvariational energies is referred to as the k(2n+1,2n+2) rule. For k=z, the well-known 2n+2 rule for the multipliers is reproduced, while the wave-function parameters satisfy the 2n+1 rule. The application of the k(2n+1) and k(2n+1,2n+2) rules minimizes the total number of response equations to be solved when the molecular property contains k extensive perturbations (e.g., geometrical derivatives) and z-k intensive perturbations (e.g., electric fields).

Electronic excitation energies are determined using the CAM-B3LYP Coulomb-attenuated functional [T. Yanai et al. Chem. Phys. Lett. 393, 51 (2004)], together with a standard generalized gradient approximation (GGA) and hybrid functional. The degree of spatial overlap between the occupied and virtual orbitals involved in an excitation is measured using a quantity Lambda, and the extent to which excitation energy errors correlate with Lambda is quantified. For a set of 59 excitations of local, Rydberg, and intramolecular charge-transfer character in 18 theoretically challenging main-group molecules, CAM-B3LYP provides by far the best overall performance; no correlation is observed between excitation energy errors and Lambda, reflecting the good quality, balanced description of all three categories of excitation. By contrast, a clear correlation is observed for the GGA and, to a lesser extent, the hybrid functional, allowing a simple diagnostic test to be proposed for judging the reliability of a general excitation from these functionals--when Lambda falls below a prescribed threshold, excitations are likely to be in very significant error. The study highlights the ambiguous nature of the term "charge transfer," providing insight into the observation that while many charge-transfer excitations are poorly described by GGA and hybrid functionals, others are accurately reproduced.

The Gaussian-type orbital and Gaussian-type geminal (GGn) model is applied to the water molecule, at the level of second-order Møller-Plesset (MP2) theory. In GGn theory, correlation factors are attached to all doubly-occupied orbital pairs (GG0), to all doubly-occupied and singly-excited pairs (GG1), or to all orbital pairs (GG2). Optimizing the GG2 model using a weak-orthogonality functional, we obtain the current best estimate of the all-electron MP2 correlation energy of water, -361.95 mE(h). In agreement with previous observations, the GG1 model performs almost as well as the GG2 model (-361.26 mE(h)), whereas the GG0 model is poorer (-351.36 mE(h)). For the barrier to linearity of water, we obtain an MP2 correlation contribution of -463 +/- 5 cm(-1).

Electronic circular dichroism (ECD) parameters of the disulphide chromophore have been calculated for dihydrogen disulphide, dimethyl disulphide, and cystine using density-functional theory, coupled-cluster theory, and multiconfigurational self-consistent field theory. The objective is twofold: first, to examine the performance of the Coulomb-attenuated CAM-B3LYP functional for the calculation of ECD spectra; second, to investigate the dependence of the ECD parameters on the conformation around the disulphide bridge. The CAM-B3LYP functional improves considerably on the B3LYP functional, giving results comparable to CCSD theory and to MCSCF theory in an extended active space. The conformational dependence of the ECD parameters does not change much upon substitution, which is promising for the application of ECD in structural investigations of proteins containing disulphide bridges.

We have implemented the use of mixed basis sets of Gaussian one- and two-electron (geminal) functions for the calculation of second-order Møller-Plesset (MP2) correlation energies. In this paper, we describe some aspects of this implementation, including different forms chosen for the pair functions. Computational results are presented for some closed-shell atoms and diatomics. Our calculations indicate that the method presented is capable of yielding highly accurate second-order correlation energies with rather modest Gaussian orbital basis sets, providing an alternative route to highly accurate wave functions. For the neon atom, the hydrogen molecule, and the hydrogen fluoride molecule, our calculations yield the most accurate MP2 energies published so far. A critical comparison is made with established MP2-R12 methods, revealing an erratic behaviour of some of these methods, even in large basis sets.

The trust-region self-consistent field (TRSCF) method is presented for optimizing the total energy E(SCF) of Hartree-Fock theory and Kohn-Sham density-functional theory. In the TRSCF method, both the Fock/Kohn-Sham matrix diagonalization step to obtain a new density matrix and the step to determine the optimal density matrix in the subspace of the density matrices of the preceding diagonalization steps have been improved. The improvements follow from the recognition that local models to E(SCF) may be introduced by carrying out a Taylor expansion of the energy about the current density matrix. At the point of expansion, the local models have the same gradient as E(SCF) but only an approximate Hessian. The local models are therefore valid only in a restricted region-the trust region-and steps can only be taken with confidence within this region. By restricting the steps of the TRSCF model to be inside the trust region, a monotonic and significant reduction of the total energy is ensured in each iteration of the TRSCF method. Examples are given where the TRSCF method converges monotonically and smoothly, but where the standard DIIS method diverges.

The basis-set convergence of cc-pVXZ basis sets is investigated for the MP2 and CCSD equilibrium bond distances and harmonic frequencies of BH, HF, CO, N-2, and F-2 by comparing with explicitly correlated R12 results. The convergence is, in general, smooth but slow-for example, for harmonic frequencies at the quadruple-zeta level, the basis-set error is typically 7 cm(-1); at the sixtuple-zeta level, it is about 2 cm(-1). For most constants, the convergence can be accelerated by using a two-point linear extrapolation procedure. Equilibrium bond distances, harmonic frequencies, anharmonic contributions, vibration-rotation interaction constants, and rotational constants for the vibrational ground state have been calculated for the same set of molecules using standard wave function and basis-set levels of ab initio theory. The accuracy of the calculated constants has been established by carrying out a statistical analysis of the deviations with respect to experiment. The largest errors for bond distances and harmonic frequencies calculated at the core-corrected CCSD(T)/cc-pV6Z level are 0.4 pm and 13.4 cm(-1), respectively. Much smaller errors occur for the anharmonic contributions. (C) 2003 American Institute of Physics.

A scheme for calculating the vibrational structure of electronic spectra using coupled-cluster response theory is proposed. To calculate the vibrational structure of electronic transitions, the optimized geometries of the two electronic states, the molecular Hessians, the dipole transition moment and (for vibrationally induced transitions) the geometrical gradient of the dipole transition moment are used in conjunction with a recently developed method for the evaluation of Franck–Condon factors of multidimensional harmonic oscillators. Allowed and vibrationally induced transitions are both described. In this pilot implementation, the required geometrical derivatives are calculated by an automated finite-difference method. The scheme is applied to the 1 1A″←1 1A′ transition of monofluorosilylene (HSiF) and the vibrationally induced 1 1A2←1 1A1 transition of water.

A reformulation of Hartree–Fock theory for time-independent molecular properties with perturbation-dependent basis sets and which refers strictly to the atomic-orbital basis is presented. The formalism is based on a recently proposed exponential parametrization of the one-electron atomic-orbital density matrix. In the presented formulation, only multiplications and additions of sparse matrices are needed. Linear scaling with system size is therefore obtainable, making this formulation ideally suited to large molecular systems. The paper contains general formulas for molecular energy derivatives up to fourth order, with special attention given to molecular gradients, molecular Hessians, magnetizabilities, and nuclear magnetic shieldings.

The distributions of the translational energy ( T) released during loss of H2 from metastable CH3NH3 + and CH3OH2 + ions have been measured. For both reactions the most probable T value accounts for approximately 3/4 of the reaction’s reverse critical energy. Subject to the same experimental conditions CH 3FH + ions do not give rise to any measurable signal for H 2 loss. The relevant parts of the potential energy surfaces of all three reactions were investigated using various ab initio quantum chemical computational schemes. Ab initio direct dynamics calculations were performed to obtain representative reaction trajectories. Translational energy releases computed at the end of these trajectories (where the fragments have separated) agree with the corresponding experimental figures. The three reactions follow a common polar mechanism which involves an initial transfer of a proton from the most basic centre (N, O or F) towards one of the hydrogen atoms of the methyl group. During this stage the proton polarizes the electrons around the methyl hydrogen to give it some hydride character, and in the transition state this has resulted in an embryonic H‐H bond. Further electron reorganization during the concerted bond breaking and bond making process leads to a strong repulsive force along the reaction coordinate as the two fragments depart. This accounts for the highly non-statistical partitioning of the available potential energy into relative translation between the two fragments formed.

The dynamic polarizability of N 2 is calculated using multiconfigurational linear response (MCLR) with restricted active space CI expansions. This allows inclusion of the static correlation and part of the dynamic correlation. The calculations which include dynamic correlation give results close to the experimental values with only a small fraction of the orbital space occupied.

The analytical calculation of molecular dipole‐moment derivatives for MCSCF wave functions is described. The formalism is based on exponential unitary transformation of the wave function and symmetric orthonormalization of the molecular orbitals. The response equations are solved using an iterative, direct technique to allow for large configuration expansions. Translational and rotational symmetries of the dipole moment are used to minimize computational costs. Sample calculations involving several thousand configurations are presented for H2O and ONF.

The calculation of molecular Hessians for large‐scale multiconfiguration self‐consistent‐field (MCSCF) functions is described. The formalism is based on exponential parametrization of the wave function and symmetric orthonormalization of the molecular orbitals. Extensive use is made of one‐index transformations of the molecular integrals, both to construct the gradient vectors that appear in the linear MCSCF response equations, and to perform the multiplication of the trial vectors on the electronic Hessian in the iterative, direct solution of the response equations. No element of the electronic Hessian is ever calculated explicitly, allowing for use of large configuration expansions. Efficient methods are developed for obtaining the solution vectors of the linear response equations. The accuracy of the molecular Hessian is analyzed in terms of the accuracy of these solution vectors. To allow for large basis sets Fock matrices are used to minimize transformations and integrals are recalculated to minimize storage requirements. Integral derivatives are calculated following the McMurchie–Davidson scheme. A simplified algorithm for calculation of derivatives of integrals involving one‐center overlap distributions is described. Sample calculations involving several thousand configurations are reported.

A Mo/ller–Plesset energy functional (Lagrangian) which is variational in all variables (the Lagrange multipliers, the orbital rotation parameters, and the orbital energies) is constructed. The variational property ensures that the responses of the orbitals and orbital energies to order n in geometrical perturbations determine the energy derivatives to order 2n+1. The Lagrange multipliers satisfy the somewhat stronger 2n+2 rule. The multipliers, orbital rotations, and orbital energy responses are determined from coupled perturbed Hartree–Fock‐type equations using an exponential parametrization of the orbitals. This ensures that the orbital rotations and energy responses are treated in the same way and calculated from a single set of linear equations. Explicit expressions for energy derivatives up to third order are derived for the second‐order Mo/ller–Plesset energy.

Gradient extremals define stream beds connecting stationary points on molecular potential energy surfaces. Using this concept we have developed an algorithm to determine transition states. We initiate walks at equilibrium geometries and follow the gradient extremals until a stationary point is reached. As an illustration we have investigated the mechanism for exchange of protons on carbon in methylenimine (H2C=NH) using a multi-reference self-consistent-field wave function.

When Moller-Plesset energy derivatives are determined in the canonical Hartree-Fock basis, singularities or instabilities may arise due to degeneracies among the occupied or unoccupied orbitals. If a non-canonical basis is used these singularities disappear. Numerically stable expressions are presented for the molecular gradient and Hessian of the second-order Moller-Plesset energy, obtained by differentiating a fully variational Lagrangian of the energy constructed in a non-canonical representation. By using a non-canonical representation, singularities and instabilities are avoided, and the variational property of the Lagrangian ensures that Wigner's 2n + 1 rule is satisfied for the orbital derivatives and that the multipliers satisfy the stronger 2n + 2 rule. It is shown that the most expensive step in the calculation of the Hessian scales as Mn 4o, where M is the number of independent Cartesian distortions, n the total number of orbitals, and o the number of occupied orbitals.

Geometrie ab initio du compose dans l'etat gazeux ou solide, simulee par ordinateur. Bon accord avec les resultats experimentaux

We derive expressions for the first through fourth derivatives of the configuration-interaction (CI) electronic energy with respect to molecular deformation. By using unitary exponential parameterizations of the wavefunction's orbital and configuration amplitude response together with a power-series expansion of the geometry dependence of the hamiltonian, a computationally attractive expression for the CI energy derivatives is obtained. The use of so-called direct methods in evaluating the CI derivatives is discussed as are the relative efforts involved in using our CI-based energy-derivative expressions and those which we obtained earlier for derivatives of the multiconfigurational self-consistent-field energy. The power-series expansion of the geometry dependence of the hamiltonian that we have derived may be used for evaluating molecular-deformation derivatives for any approximate wavefunction constructed from a set of orthonormal orbitals.

Compact expressions are derived for the analytical gradient with a general MC SCF LCAO wavefunction. Cancellations among the terms involved are shown to have important computational aspects, and substantial reductions of computer time may be achieved with a properly chosen basis set.

We present theory and implementation for a new approach for studying solvent effects: the multiconfigurational self‐consistent reaction‐field (MCSCRF) method. The atom, molecule, or supermolecule is assumed to be surrounded by a linear, homogeneous, continuous medium described by its macroscopic dielectric constant. The electronic structure of the compound is described by a multiconfigurational self‐consistent field (MCSCF) wave function. The wave function is fully optimized with respect to all variational parameters in the presence of the surrounding polarizable dielectric medium. We develop a second‐order convergent optimization procedure for the solvent states. The solvent integrals are evaluated by an efficient and general algorithm. The flexible description of the electronic structure allows us to accurately describe ground, excited, or ionized states of the solute. Deficiencies in the calculation can therefore be assigned to the cavity model rather than the description of the solute.

A configuration-interaction energy function (Lagrange) which is variational in all variables, including the orbital rotational parameters, is constructed. When this Lagrangian is used for obtaining configuration-interaction derivatives, all the important simplifications which occur for derivatives of variational wave functions carry over in a straightforward way. In particular, the state and orbital rotational response parameters obey the 2n+1 rule and the Lagrange multipliers obey the somewhat stronger 2n+2 rule. The simplifications which are normally obtained by invoking the Handy-Schaefer technique are automatically incorporated to all orders. Simple expressions for energy derivatives up to third order are presented. The relationship between the numerical errors in the variational parameters and the errors in the calculated energy derivatives is discussed.

The calculation of molecular electronic wave functions and properties using floating Gaussian orbitals (i.e., orbitals whose positions are optimized in space) is described. The wave function is optimized using a second‐order convergent scheme (the trust‐region method), and molecular properties up to second order are calculated analytically. The method is applied to a series of small molecules (HF, H2O, NH3, CH4, CO, H2CO, and C2H4) at the Hartree–Fock level using four different floating basis sets (double zeta, double zeta plus polarization, double zeta plus diffuse, and double zeta plus polarization and diffuse). Geometries are fully optimized, and dipole moments, static polarizabilities, harmonic frequencies, and double‐harmonic infrared intensities are calculated at the optimized geometries. The results are compared with those obtained using the corresponding fixed basis sets, and also with the results from a large basis of near‐Hartree–Fock quality [6–311++G(3df,3pd)]. Floating produces only minor cha...

A general theory for response properties is presented which is applicable to perturbations affecting the molecular basis set, notably nuclear derivatives. A perturbation-independent Fock space is introduced, and the necessary reorthonormalization of a truncated basis set after a perturbation is explicitly incorporated in the Hamiltonian. Explicit formulas for MCSCF first- and second-order properties are presented, and some computational aspects are briefly discussed. A brief comparison with previous results is given.

The restricted step optimization algorithm is applied to potential energy surfaces calculated from multiconfiguration self‐consistent‐field wave functions. Equilibrium and transition‐state geometries are determined by iteratively solving a set of level‐shifted Newton–Raphson equations. At each geometry the molecular gradient and Hessian are calculated analytically, and a first‐order prediction of the wave function at the next geometry is obtained by combining the geometrical derivatives of the wave function with the geometrical step vector. The usefulness of this prediction is discussed and illustrated by test calculations. The numerical accuracy which is required in the wave function and its geometrical derivatives in order to maintain quadratic convergence in the optimization of the molecular geometry is analyzed. It is demonstrated that the Newton–Raphson step vector and the wave function prediction may be determined without calculating the molecular Hessian explicitly. Sample calculations are carried ...

We present a Gaussian-basis implementation of orbital-free density-functional theory (OF-DFT) in which the trust-region image method (TRIM) is used for optimization. This second-order optimization scheme has been constructed to provide benchmark all-electron results with very tight convergence of the particle-number constraint, associated chemical potential, and electron density. It is demonstrated that, by preserving the saddle-point nature of the optimization and simultaneously optimizing the density and chemical potential, an order of magnitude reduction in the number of iterations required for convergence is obtained. The approach is compared and contrasted with a new implementation of the nested optimization scheme put forward by Chan, Cohen, and Handy. Our implementation allows for semilocal kinetic-energy (and exchange-correlation) functionals to be handled self-consistently in all-electron calculations. The all-electron Gaussian-basis setting for these calculations will enable direct comparison with a wide range of standard high-accuracy quantum-chemical methods as well as with Kohn-Sham density-functional theory. We expect that the present implementation will provide a useful tool for analyzing the performance of approximate kinetic-energy functionals in finite systems.

Publisher Summary This chapter reviews the techniques recently available for the analytical calculation of molecular energy and property derivatives. The chapter focuses on the techniques that are simple and general, in particular those which can handle large basis sets and configuration expansions. It presents derivative expressions for most commonly used methods in molecular electronic structure theory, and their computational implementations. It is seldom possible to give expressions that are optimal under all conditions, regardless of the number of electrons, basis functions, and configurations. The analytical calculation of energy derivatives has recently been reviewed by Pulay and Gaw and Handy and the analytical calculation of property derivatives by Amos. The development of and computational implementation of analytical expressions for the derivatives of ab initio electronic energy surfaces and molecular properties have undergone rapid growth in recent years. This growth reflects the central role these derivatives play in understanding chemical reactions and in interpreting many spectroscopic experiments. The approach taken in this chapter constitutes a good basis for developing efficient procedures for calculation of geometrical derivatives under any particular set of circumstances.

The ab initio calculation of molecular dipole moments, polarizabilities and their geometrical derivatives is discussed. General expressions valid for variational wave functions (SCF, MCSCF and CI) and perturbation-dependent basis sets are derived and their implementation is discussed. The use of direct methods to simplify the calculation of higher-order properties is described.

In this paper we consider the geometry dependence of the molecular Hamiltonian. The Hamiltonian is constructed in a geometry-dependent orbital representation. Orbital connections are introduced to link such representations at different geometries, and it is shown how orthogonal connections lead to geometry-independent density elements. Derivative expressions of the Hamiltonian are given in terms of one-index transformations of the integrals. It is illustrated how the Hellmann-Feynman theorem may be applied directly to SCF and limited CI wave functions once the right orbital connections have been chosen. Some computational aspects are considered and finally the relation to covariant derivatives in differential geometry is discussed.

Recently, we have implemented a scheme for the calculation of the adiabatic connection linking the Kohn–Sham system to the physical, interacting system. This scheme uses a generalized Lieb functional, in which the electronic interaction strength is varied in a simple linear fashion, keeping the potential or the density fixed in the process. In the present work, we generalize this scheme further to accommodate arbitrary two-electron operators, allowing the calculation of adiabatic connections following alternative paths as outlined by Yang [J. Chem. Phys. 109, 10107 (1998)] . Specifically, we examine the error-function and Gaussian-attenuated error-function adiabatic connections. It is shown that while the error-function connection displays some promising features, making it amenable to the possible development of new exchange-correlation functionals by modeling the adiabatic connection integrand, the Gaussian-attenuated error-function connection is less promising. We explore the high-density and strong static correlation regimes for two-electron systems. Implications of this work for the utility of range-separated schemes are discussed.

Molecular magnetizabilities have been calculated at the Hartree–Fock level for a series of diamagnetic molecules: H2O, NH3, CH4, PH3, H2S, CO2, CSO, CS2, and C3H4. Gauge invariance is imposed by the use of London atomic orbitals. The results are compared to those obtained with the IGLO (individual gauge for localized orbitals) method and are found to converge faster to the basis set limit. Magnetizabilities obtained from basis sets of different quality never differ by more than 4% for the London method, compared to 20% for IGLO. The Hartree–Fock limit may be approached using London basis sets of moderate size, in contrast to calculations of molecular polarizabilities which require large basis sets to be reliable. Comparison with experiment shows that the Hartree–Fock method overestimates experimental susceptibilities by 5%–10%.

We apply a recently developed model for calculating gauge origin independent molecular magnetic properties of solvated molecules to determine the magnetizability and nuclear shielding constants of solvated water. The solute is surrounded by a solvation shell and this supermolecule is enclosed in a spherical cavity immersed in a dielectric medium.

Using multiconfigurational self-consistent field (MCSCF) wave functions and perturbation-dependent basis sets, the effect of electron correlation on molecular magnetizabilities and rotational g tensors is investigated. The eight molecular systems considered (H2O, NH3, HF, C2H2, CO, H2CO, O3, and LiH) vary in the importance and relative magnitudes of the static and dynamic correlation contributions. The results for O3 are the first correlated calculations of the rotational g tensor of this system. We confirm previous findings that, except for systems with large static correlation effects, the effect of correlation on molecular magnetizabilities is small. A somewhat larger correlation contribution is usually observed for the rotational g tensor, although this property is also rather insensitive to the correlation treatment. Agreement with experimental rotational g tensors is only fair and estimates of rovibrational corrections are needed to assess properly the accuracy of theoretically calculated rotational...

We present full CI results for the magnetizability and rotational g factor of the hydrogen molecule. Convergence of the results with respect to the size of the one-particle basis has been considered, as well as corrections for rovibrational effects. Our results at T = 300 K are: (−67.13 ± 0.04)·10−30 JT−2 for the isotropic magnetizability, (8.91 ± 0.03)·10−30 JT−2 for the magnetizability anisotropy and 0.8817 ± 0.0002 for the rotational g factor. The results are compared with molecular beam experiments and other theoretical results.

The static hypermagnetizabilities and nuclear shielding polarizabilities of the carbon and hydrogen atoms of ethylene have been computed using multiconfigurational linear-response theory and a finite-field method, in a mixed analytical-numerical approach. Extended sets of magnetic-field-dependent basis functions have been employed in large MCSCF calculations, involving active spaces giving rise to a few million configurations in the finite-field perturbed symmetry. The convergence of the observables with respect to the extension of the basis set as well as the effect of electron correlation have been investigated. Whereas for the shielding polarizabilities we can compare with other published SCF results, the ab initio estimates for the static hypermagnetizabilities and the observable to which they are related - the Cotton-Mouton constant, - are presented for the first time.

Ab initio calculations have been performed for the static and dynamic polarizability and hyperpolarizability for a series of polyynes C2nH2 using self‐consistent field (SCF) (n=1–6) and multiconfiguration self‐consistent field (MCSCF) (n=1–4) wave functions. We have considered the longitudinal component of the static hyperpolarizability and the same component of the dynamic hyperpolarizability measured in electric‐field induced second harmonic generation γESHG=γ(−2ω;ω,ω,0). The frequency dependence of the polarizability and hyperpolarizability has been rationalized in terms of the coefficients in expansions in ω2. The static hyperpolarizabilities vary smoothly with the chain length and satisfy γ(C2nH2)=γ (C2H2)×nX, where X≊3.0. The dynamic hyperpolarizability satisfies a similar relation where X increases slowly with ω.