A hybrid approach of the multi-conÐguration time-dependent Hartree and Ðlter-diagonalisation methods for computing bound-state spectra. Application to HO 2 F. Gatti, M. H. Beck, G. A. Worth and H.-D. Meyer* T heoretische Chemie, Physikalisch-Chemisches Institut, Heidelberg, Im Neuenheimer Universita  t Feld 229, D-69120 Heidelberg, Germany. E-mail : Hans-Dieter.Meyer=tc.pci.uni-heidelberg.de Received 13th December 2000, Accepted 20th February 2001 First published as an Advance Article on the web 27th March 2001 A hybrid approach of the multi-conÐguration time-dependent Hartree (MCTDH) and the Ðlter-diagonalisation (FD) methods for computing bound-state spectra is applied to the study of the HO 2 radical. We investigate the efficiency and accuracy of this approach for the case where the potential energy surface is not given as a sum of products of one-dimensional functions. As the MCTDH scheme requires such a product form in order to be efficient the potential energy surface was replaced by a potential Ðt of the required form. The performance of our approach is compared with that of the Lanczos algorithm. 1 Introduction The study of excited states of bound systems is a very impor- tant topic in chemical physics since it concerns all spectros- copy. In order to obtain the eigenstates of bound systems several strategies have been proposed. For instance, the Lanczos algorithm1 is an efficient time-independent approach to determine the desired eigenenergies but it shows fast con- vergence only at the opposite ends of the spectrum and is slow for resolving the important interior region. Time-dependent techniques became competitive for such systems when Neu- hauser proposed his Ðlter-diagonalization (FD) method.2 The Ðlter-diagonalization method allows one to obtain efficiently the eigenvalues at the bottom of the spectrum as well as in the interior region. Very recently a fast and sufficiently accurate time-dependent approach for determining bound-state spectra has been suggested by two of us.3 This hybrid approach exploits the efficiency of the multi-conÐguration time- dependent Hartree (MCTDH) scheme for propagating wavepackets4 h8 to produce an auto-correlation function. From this, an accurate spectrum is extracted employing the Ðlter-diagonalisation (FD) method.2,9 h16 In a previous calcu- lation, this MCTDH/FD approach was applied to the compu- tation of the vibrational spectrum of For this system it CO 2 . was shown that the MCTDH/FD method is superior in effi- ciency not only to the FD scheme based on a numerically exact propagation, but also to the Lanczos algorithm.3 However, some questions have remained open which will be addressed in this article. The Ðrst is whether a similar efficiency and accuracy can be achieved also for other systems. We therefore apply the MCTDH/FD method to the calculation of the bound states of a very di†erent system, namely the radical. Our results HO 2 are also compared with those obtained with the Lanczos algo- rithm.1,17,18 The second question concerns the representation of the potential energy surface. For full efficiency, the MCTDH algo- rithm requires the potential to be given as a sum of products of one-dimensional functions. (Alternatively one may use the correlated discrete variable representation CDVR19 which employs single-particle functions to generate a time-dependent grid). While the potential we employed in our previous study on the spectrum had this product form, this is not the CO 2 case for the potential used here. Although there exists an HO 2 algorithm for constructing a potential Ðt of the required form with arbitrary accuracy,8,20,21 the use of such a Ðt might reduce the efficiency of the MCTDH scheme, as the numerical e†ort for the propagation increases linearly with the number of product terms. We will hence investigate the performance of the MCTDH/FD method when a potential Ðt is employed. The third question refers to the energy window studied. In ref. 3 we focused on the low-energy range containing the Ðrst Ðfty eigenstates. In this work, we will also examine energy windows which lie higher in energy. 2 Method Before we present our results, we will brieÑy describe the methods we have used in our calculations, namely the MCTDH and the FD scheme. For a comprehensive dis- cussion we refer the reader to ref. 4, 5 and 8 and 9, 13 and 14, respectively. 2.1 The multi-conÐguration time-dependent Hartree method We assume that the system under consideration has f degrees of freedom described by nuclear coordinates ..., In Q 1 , Q f . order to represent the Hamiltonian operator, we start from the following primary direct N 1 ] ÉÉÉ ] N f -dimensional product basis set where the are Ms i1 (1)(Q 1 )ÉÉÉs if (f)(Q f )N, s ii (i) usually chosen as the basis functions of a discrete variable rep- resentation (DVR).22h25 The Ðrst problem faced in large-scale quantum mechanical studies concerns the huge dimension of the primitive basis set. In order to reduce the size of the quantum mechanical problem, a set of contracted time-dependent functions t)T known as single-particle functions, is deÐned by o r ji (i)(Q i , means of the MCTDH algorithm in the following way. The single-particle functions are expanded in the set of the primi- tive basis functions, o r ji (i)(Q i , t)T \ ; ii/1 Ni c iiji (i) (t) o s ii (i)(Q i )T, (1) with i \ 1, . . . , f and ..., j i \ 1, n i . 1576 Phys. Chem. Chem. Phys., 2001, 3, 1576È1582 DOI : 10.1039/b009949j This journal is The Owner Societies 2001 (