In the last three decades, we have witnessed incredible advances in laser technology and in the understanding of nonlinear laser-matter interactions, crowned recently by the award of the Nobel prize to Gérard Mourou and Donna Strickland [1,2]. It is now routinely possible to produce few-cycle femtosecond (1 fs = 10 −15 s) laser pulses in the visible and mid-infrared regimes [3,4]. By focusing such ultrashort laser pulses on gas or solid targets, possibly in a presence of nano-structures [5], the targets are subjected to an ultra-intense electric field, with peak field strengths approaching the binding field inside the atoms themselves. Such fields permit the exploration of the interaction between strong electromagnetic coherent radiation and an atomic or molecular system with unprecedented spatial and temporal resolution [6]. On one hand, HHG nowadays can be used to generate attosecond pulses in the extreme ultraviolet [7,8], or even in the soft X-ray regime [9]. Such pulses themselves may be used for dynamical spectroscopy of matter; despite carrying modest pulse energies, they exhibit excellent coherence properties [10,11]. Combined with femtosecond pulses they can also be used to extract information about the laser pulse electric field itself [12]. HHG sources therefore offer an important alternative to other sources of XUV and X-ray radiation: synchrotrons, free electron lasers, X-ray lasers, and laser plasma sources. Moreover, HHG pulses can provide information about the structure of the target atom, molecule or solid [13][14][15]. Of course, to decode such information from a highly nonlinear HHG signal is a challenge, and that is why a possibly perfect, and possibly "as analytical as possible" theoretical understanding of these processes is in high demand. Here is the first instance where SFA offers its basic services.Since electronic motion is governed by the waveform of the laser electric field, an important quantity to describe the electric field shape is the so-called absolute phase or carrier-envelope phase (CEP). Control over the CEP is paramount for extracting information about electron dynamics, and to retrieve structural information from atoms and molecules [13,16,17]. For instance, in HHG an electron is liberated from an atom or molecule through ionization, which occurs close to the maximum of the electric field. Within the oscillating field, the electron can thus accelerate along oscillating trajectories, which may result in recollision with the parent ion, roughly when the laser field approaches a zero value. Control over the CEP is particularly important for HHG, when targets are driven by laser pulses comprising only one or two optical cycles. In that situation the CEP determines the relevant electron trajectories, i.e. the CEP determines whether emission results in a single or in multiple attosecond bursts of radiation [16,18].The influence of the CEP on electron emission is also extremely important. It was demonstrated for instance in an anti-correlation experiment, in which the number of AT...

The exchange splitting J of the interaction energy of the hydrogen atom with a proton is calculated using the conventional surface-integral formula J surf [ϕ], the volume-integral formula of the symmetry-adapted perturbation theory J SAPT [ϕ], and a variational volume-integral formula J var [ϕ]. The calculations are based on the multipole expansion of the wave function ϕ, which is divergent for any internuclear distance R. Nevertheless, the resulting approximations to the leading coefficient j 0 in the large-R asymptotic series J(R) = 2e −R−1 R(j 0 + j 1 R −1 + j 2 R −2 + · · · ) converge, with the rate corresponding to the convergence radii equal to 4, 2, and 1 when the J var [ϕ], J surf [ϕ], and J SAPT [ϕ] formulas are used, respectively. Additionally, we observe that also the higher j k coefficients are predicted correctly when the multipole expansion is used in the J var [ϕ] and J surf [ϕ] formulas. The SAPT formula J SAPT [ϕ] predicts correctly only the first two coefficients, j 0 and j 1 , gives a wrong value of j 2 , and diverges for higher j n . Since the variational volume-integral formula can be easily generalized to many-electron systems and evaluated with standard basis-set techniques of quantum chemistry, it provides an alternative for the determination of the exchange splitting and the exchange contribution of the interaction potential in general.

The exchange-splitting energy J of the lowest gerade and ungerade states of the H 2 + molecular ion was calculated using a volume integral expression of symmetry-adapted perturbation theory and standard basis set techniques of quantum chemistry. The performance of the proposed expression was compared to the well-known surface-integral formula. Both formulas involve the primitive function, which we calculated employing either the Hirschfelder-Silbey perturbation theory or the conventional Rayleigh-Schrödinger perturbation theory (the polarization expansion). Our calculations show that very accurate values of J can be obtained using the proposed volume-integral formula. When the Hirschfelder-Silbey primitive function is used in both formulas the volume formula gives much more accurate results than the surface-integral expression. We also show that using the volume-integral formula with the primitive function approximated by Rayleigh-Schrödinger perturbation theory, one correctly obtains only the first four terms in the asymptotic expansion of the exchange-splitting energy.

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The exchange contribution to the energy of the hydrogen atom interacting with a proton is calculated from the polarization expansion of the wave function using the conventional surfaceintegral formula and two formulas involving volume integrals: the formula of the symmetry-adapted perturbation theory (SAPT) and the variational formula recommended by us. At large internuclear distances R, all three formulas yield the correct expression −(2/e)Re −R , but approximate it with very different convergence rates. In the case of the SAPT formula, the convergence is geometric with the error falling as 3 −K , where K is the order of the applied polarization expansion. The error of the surface-integral formula decreases exponentially as a K /(K + 1)!, where a = ln 2 − 1 2. The variational formula performs best, its error decays as2 . These convergence rates are much faster than those resulting from approximating the wave function through the multipole expansion. This shows the efficiency of the partial resummation of the multipole series effected by the polarization expansion. Our results demonstrate also the benefits of incorporating the variational principle into the perturbation theory of molecular interactions.It is impossible to understand the world without the knowledge of intermolecular interactions [1]. Not only do they govern the properties of gases [2], liquids [3], and solids [4], but also influence chemical reactivity [5] and determine the structure of complex biological systems [6].The most straightforward perturbation treatment of molecular interactions, known as the polarization approximation [7] or polarization expansion, consists in an application of the standard Rayleigh-Schrödinger perturbation theory, with the zeroth-order Hamiltonian H 0 taken as the sum of the non-interacting monomer Hamiltonians, and the perturbation V (the interaction operator) defined as V = H − H 0 , where H is the electronic Hamiltonian of the system. Polarization expansion provides the correct, valid for all intermolecular distances R, definitions of the electrostatic, induction, and dispersion contributions to the interaction energy [8]. It is well known, however, that in a practically computable finite order, the polarization expansion for the energy is not able to recover the exchange energy, the basic repulsive component of the interaction potential that determines the structure of molecular complexes and solids. It is also known [9,10] that the polarization series provides the asymptotic expansion of the primitive functionwhere ϕ (k) is the kth-order (in V ) polarization correction to the wave function and κ = 3 for interactions of neutral monomers, and κ = 2 when at least one of the monomers is charged. Equation (1) represents the genuine primitive function in the sense of Kutzelnigg [11], i.e., the function which, after appropriate symmetry projections A ν , yields correctly all asymptotically degenerate wave functions Ψ ν of the interacting system, A ν Φ = Ψ ν , and which is localized in the same way as the zeroth-order w...