Ab-initio complex molecular potential energy surfaces by the back-rotation transformation method P. Balanarayan a , Y. Sajeev b , Nimrod Moiseyev a,⇑ a Schulich Faculty of Chemistry and Minerva Center of Nonlinear Physics of Complex Systems, Technion-Israel Institute of Technology, Haifa 32000, Israel b Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Heidelberg D-69120, Germany article info Article history: Received 14 October 2011 In final form 13 December 2011 Available online 19 December 2011 abstract The imaginary part of the complex potential energy surface (CPES) provides the autoionization, Auger and ICD (Intermolecular Coulombic Decay) ionization decay rates as functions of the variations in the molec- ular electronic structure and geometry. We introduce here a novel approach which enables the calcula- tions the molecular complex potential energy surfaces by the standard computational algorithms which were originally developed for calculating real PES of stable molecules. As an illustrative numerical exam- ple we have calculated the CPES for the molecular autoionization of hydrogen molecule, H 2 ½ 1 R þ g  where the two electrons are doubly excited. Ó 2011 Elsevier B.V. All rights reserved. 1. Motivation and the problem The recent developments in free-electron laser technology and synchrotron radiation sources provide the facilities needed to carry out experiments on electronic wave packet dynamics. Theoretical studies of electronic wave packet dynamics in molecules is a chal- lenge mainly because of the need to include couplings between the electronic and nuclear coordinates into molecular wave packet propagation calculations. These couplings can have a dramatic ef- fect on the molecular dynamics. Apart from this problem, it is also of interest to analyze within the adiabatic approximation the effect of molecular ionization as a function of nuclear coordinates. For example, it might happen that the motion of the electronic wave packet is accomplished via molecular ionization. The rate of ioniza- tion depends heavily on nuclear motions. However, molecular propagation calculations where the elec- tron–nuclei couplings (non adiabatic effects) are taken into consid- eration are very difficult, if not impossible, even with the most advanced computational facilities. In order to include electronic ionization effects while the nuclei are moving (within the adiabatic approximation), the time-depen- dent Schrödinger equation must be solved with complex potential energy surfaces (CPES) [1–3]. The CPES are the complex eigenvalues of the electronic Born– Oppenheimer (BO) molecular Hamiltonian as functions of the nu- clear positions, fR a g. CPES are also obtained with complex absorb- ing potentials (CAPs) added to impose outgoing boundary conditions [5]. In order to minimize the artificial reflection effects on the molecular dynamics which are due to the CAPs, the CAP parameters should be optimized. There are no universal problem independent CAPs. Hence calculating a reliable complex PES is still an open problem [6]. In principle we know that we need to solve the electronic time independent Schrödinger equation within the BO approximation with outgoing boundary conditions (so called Siegert boundary conditions) that describe the free-moving ionized electrons. Under outgoing boundary conditions the BO Hamilto- nian has a discrete electronic spectrum, with real discrete eigen- values associated with electronic bound states and complex discrete eigenvalues associated with metastable (resonance) elec- tronic states, E n ðfR a gÞ, where 2Im½E n ðfR a gÞ is the ionization de- cay rate, b H elec W elec n ðfr j g j¼1;... ; fR a g a¼1;2;... Þ¼ E n ðfR a g a¼1;... ÞW elec n ðfr j g j¼1;... ; fR a g a¼1;2;... Þ lim jr j j!1 W elec n ðfr j g j¼1;...N ; fR a g a¼1;2;... Þ!1 b H elec ¼ T e þ V eN þ V ee ð1Þ Here T e , V eN and V ee are respectively, the electronic kinetic energy operator, electron–nuclei potential energy operator and the electronic repulsion energy operator. In Eq. (1) the wavefunction tends to infinity resulting from the requirement of outgoing boundary condition. This behavior results from the fact that the outgoing wavefunction behaves like exp(ikr) asymptotically. For bound states k ¼ ijkj and for resonances k ¼jkjr ia such that expðikrÞ!1. By imposing outgoing (Siegert) boundary conditions a discrete spectra is obtained associated with bound (real eigen- value) and resonance /complex eigenvalues) states. 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.12.028 ⇑ Corresponding author. E-mail address: nimrod@technion.ac.il (N. Moiseyev). Chemical Physics Letters 524 (2012) 84–89 Contents lists available at SciVerse ScienceDirect Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett