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Physica A 284 (2000) 131–139 www.elsevier.com/locate/physa An evolutionary algorithm to calculate the ground state of a quantum system a Institut I. Grigorenkoa; ∗ , M.E. Garciab fur Theoretische Physik, Freie Universitat Berlin, Arnimallee 14, 14195 Berlin, Germany de FÃsica Teorica, Universidad de Valladolid, 47011 Valladolid, Spain b Departamento Received 14 March 2000 Abstract We present a new method based on evolutionary algorithms which permits to determine efciently the ground state of the time-independent Schrodinger equation for arbitrary external potentials. The approach relies on the variational principle. The ground-state wave function of a given Hamiltonian is found by using the procedure of survival of the ttest, starting from a population of wave functions. To perform the search for the ttest wave function we have extended a genetic algorithm to treat quantum mechanical problems. We present results for dierent one dimensional external potentials and compare them with analytical solutions and with other numerical methods. Our approach yields very good convergence in all cases. Potential applications of the quantum genetic algorithm presented here to more dimensions and many-body problems c 2000 Elsevier Science B.V. All rights reserved. are discussed. PACS: 02.90.+p; 03.65.−w Keywords: Genetic algorithms; Quantum mechanics 1. Introduction In the last years an increasing number of studies of the ground-state of electronic systems in real space has been performed [1–7]. The usefulness of numerical calculations in coordinate space has also been recently demonstrated [8]. The real space calculations performed until now can be divided into two dierent groups. The rst one deals with the expansion of the wave function of the system in terms of basis functions (e.g. eigenstates of the angular momentum or of the harmonic oscillator). This method is restricted to highly symmetric boundary conditions (circular, ∗ Corresponding author. E-mail addresses: [email protected] (I. Grigorenko), [email protected] (M.E. Garcia) c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 2 1 8 - 1 132 I. Grigorenko, M.E. Garcia / Physica A 284 (2000) 131–139 rectangular, spherical, etc.) [3,6]. Otherwise the Hamiltonian matrix is no longer sparse and its numerical diagonalization becomes dicult. A less traditional approach consists in the discretization of the space, representing it by a lattice having a given number of mesh points. This method can be interpreted as an expansion of the wave function of the system in terms of position eigenstates of the discretized space. For the determination of the ground state using the mesh two dierent procedures have been used so far. Some authors write the Hamiltonian on the local basis and diagonalize it [5]. This would, in principle, allow to consider external potentials of arbitrary symmetry. However, nite-size or border eects can be important if the symmetry of the mesh is dierent from that of the system. In order to minimize these eects the number of lattice points has to be increased, which leads to large matrices and the consequent numerical diculties. An alternative to the diagonalization is the imaginary-time propagation [1,2,7] to determine the ground state, starting from an initial wave function written in the discretized space. This method, whose convergence is very sensitive to the time-step of the evolution [9], seems to be more suitable for low-symmetry potentials. However, the x-point of the imaginary-time propagation is not necessarily the true ground state of the system, but it could be a local minimum of the energy hyper-surface. This is a very common problem in the search for the global minimum of a complicated function or functional. In general, constrained search procedures like imaginary-time propagation frequently become trapped in a local minimum. The probability of trapping can be reduced, to some extent, by introducing a certain degree of randomness or noise (and in fact this can be achieved by increasing the time-step of the propagation [9]). However, random searches are not ecient for problems involving complex hyper-surfaces, as is the case of the ground-state system of a system under the action of a complicated external potential. In this paper we present a completely dierent and unconventional approach for the calculation of the wave function and the energy of the ground states of quantum systems, which is based on a genetic algorithm, a technique which resembles the process of evolution in nature. The genetic algorithm (GA) belongs to a new generation of the so-called intelligent global optimization techniques. First proposed by Holland [10,11] in connection with his theory of adaptive systems, it has been applied to numerous dicult problems of optimization, particularly in engineering and applied sciences (see, for instance, Refs. [12,13] for a review on GA). The GA handles problems even in highly nonlinear, multidimensional search spaces with surprising speed and eciency. This search method has been recently applied to optimize the atomic structures of small clusters [14 –19]. In these works the global minimum of the cohesive energy was obtained for dierent cluster species using Lennard–Jones potentials [14], ionic potentials [17], or interaction potentials derived from tight-binding Hamiltonians [15,16,19]. In all extensions or applications of evolutionary algorithms performed up to now, including the above mentioned structure optimizations, only classical objects have been I. Grigorenko, M.E. Garcia / Physica A 284 (2000) 131–139 133 treated. In this paper we present the rst extension of genetic algorithms to the quantum world. The paper is organized as follows. In Section 2 we outline the method and the extension of the algorithm to account for quantum mechanical problems. In Section 3 we present results for the ground state wave function corresponding to dierent onedimensional potentials. Results are compared with analytical calculations. Finally, in Section 4 we summarize our results and discuss the possible applications of our method to more complicated problems in higher dimensions. 2. Theory Before describing our approach we present rst a brief description of the GA. As we have mentioned before, the GA was developed to optimize (maximize or minimize) a given property, like an area, a volume or an energy. The property in question is a function of many variables of the system. In GA-language this quantity is referred as the tness function. There are many dierent ways to apply the GA. One of them is the phenotype version. In this approach, the GA basically maps the degrees of freedom or variables of the system to be optimized onto a genetic code (represented by a vector). Thus, a random population of individuals is created as a rst generation. This population “evolves” and subsequent generations are reproduced from previous generations through application of dierent operators on the genetic codes, like, for instance, mutations, crossovers and reproductions or copies. The mutation operator changes randomly the genetic information of an individual, i.e., one or many components of the vector representing its genetic code. The crossover or recombination operator interchanges the components of the genetic codes of two individuals [13]. In a simple recombination, a random position is chosen at which each partner in a particular pair is divided into two pieces. Each vector then exchanges a section of itself with its partner. The copy or reproduction operator merely transfers the information of the parent to an individual of the next generation without any changes. In our present approach the vector representing the genetic code is just the wave function (x). The tness function, i.e., the function to be optimized by the successive generations is the expectation value E[ ] = h |Ĥ i=h | i ; (1) where the 1D-Hamiltonian is given by Ĥ = − 12 ∇2 + V (x) : (2) Here, V (x) is the external potential. There are many dierent ways to describe the evolution of the population and the creation of the osprings. The genetic algorithm we propose to obtain the ground state of a quantum system can be described as follows: (i) We create a random initial population { j(0) (x)} consisting of N wave functions. (ii) The tness E[ j(0) ] of all individuals is determined. (iii) A new population { j(1) (x)} is created through application 134 I. Grigorenko, M.E. Garcia / Physica A 284 (2000) 131–139 of the genetic operators. (iv) The tness of the new generation is evaluated. (v) Steps (iii) and (iv) are repeated for the successive generations { j(n) (x)} until convergence is achieved and the ground-state wave function is found. Usually, real-space calculations deal with boundary conditions on a box. Therefore, and in order to describe a wave function within a given interval a6x6b we have to choose boundary conditions for (a) and (b). For simplicity we set (a) = (b) = 0, i.e., we consider a nite box with innite walls at x = a and x = b. Inside this box we can simulate dierent kinds of potentials, and if the size of the box is large enough boundary eects on the results of our calculations can be minimized. As initial population of wave functions satisfying the boundary conditions j (a) = 0, j (b) = 0 we choose Gaussian-like functions of the form j (x) = A exp(−(x − xj )2 =j2 )(x − a)(b − x) ; (3) with random values for xj ∈ [a; b] and j ∈ R(0; b − a], whereas the amplitude A is calculated from the normalization condition | (x)|2 d x = 1 for given values of xj and j . As we have mentioned above, we dene three kinds of operations on the individuals: reproduction and mutation of a function, and crossover between two functions. While the reproduction operation has the same meaning as in previous applications of the GA, both the crossover and the mutation operations have to be redened to be applied to the quantum mechanical case. The smooth or “uncertain” crossover is dened as follows. Let us take two randomly chosen “parent” functions 1(n) (x) and 2(n) (x). We can construct two new functions 1(n+1) (x), 2(n+1) (x) as (n+1) (x) = 1 (n+1) (x) = 2 (n) 1 (x) St(x) (n) 2 (x) St(x) (n) 2 (x) (1 (n) 1 (x) (1 + + − St(x)) ; − St(x)) ; where St(x) is a smooth step function involved in the crossover operation. We consider St(x) = (1 + tanh((x − x0 )=kc2 ))=2, where x0 is chosen randomly (x0 ∈ (a; b)) and kc is a parameter which allows to control the sharpness of the crossover operation. The idea behind the “uncertain” crossover is to avoid large derivatives of the new generated wave functions. Note, that the crossover operation between identical wave functions generates the same wave functions. The mutation operation in the quantum case must also take into account the uncertainty relations. It is not possible to change randomly the value of the wave function at a given point without producing dramatic changes in the kinetic energy of the state. To avoid this problem we dene the mutation operation as (n+1) (x) = (n) (x) + r (x) ; (4) where r (x) is the random mutation function. In the present case we choose r (x) as a Gaussian r (x) = B exp (−(xr − x)2 =R2 ) with a random center xr ∈ (a; b), width R ∈ (0; b − a), and amplitude B. For each step of GA iteration we randomly perform copy, crossover and mutation operations. After the application of the genetic operation the newly created functions are normalized. In the next section we present our results. I. Grigorenko, M.E. Garcia / Physica A 284 (2000) 131–139 135 Fig. 1. Ground-state spatial density distribution of the electron | (x)|2 in a one-dimensionalRinnite well (de∗ (x)Ĥ (x) d x ned in the interval [0; 1]) calculated using GA. Inset gure: evolution of the tness E[ ]= as a function of the number of iterations. 3. Results and conclusions We have performed calculations of the ground state wave function (x) for dierent potentials in one dimension. We have used Pm = 0:03 for the probability of a mutation and Pc = 0:97 for the probability of a crossover operation. For each iteration we evaluateR the tness function for the dierent individuals of ∗ the population as Ej = E[ j ] = j (x)Ĥ j (x) d x and follow the steps described in the previous section. This process is repeated until the values of the tness function converge to the minimal value of the energy. We consider dierent types of external potentials V (x) to demonstrate the eciency of the algorithm. In all examples presented here we represented the space by a lattice with 300 grid points. In the gures presented below we show the results for the density probability of the ground state and the behavior of the tness function during the iterative GA-procedure. In Fig. 1 we show the calculated ground state particle density | (x)|2 for a potential well with innite walls at x = 0 and x = 1. The ground-state energy calculated using our method is E0 = 4:9335, while the exact value is E0 = 2 =2 = 4:9348 (throughout this paper we use atomic units). performed calculations for the harmonic potential Vh (x) = 12 !2 x2 , with ! = √ We also 20×102 . In Fig. 2 the calculated ground density state is shown. It lies so close to the analytical solution, that the discrepancies could not be distinguished any more in the 136 I. Grigorenko, M.E. Garcia / Physica A 284 (2000) 131–139 Fig. 2. Calculated spatial distribution of electron density | (x)|2 (solid line) for the 1D harmonic potential (dotted line). The inset gure shows the evolution of the tness during the GA-iterations. Fig. 3. Calculated spatial distribution of the electron density | (x)|2 (solid line) for an anharmonic potential of the fourth order (dotted line). The inset gure shows the evolution of the tness as a function of the number of iterations. plot. For the ground-state energy the GA yields E0 = 316:29, while the analytical result is E0 = 316:22, which represents a discrepancy of less than 0:02% after 100 iterations. In Fig. 3 we present the calculated ground-state density for an anharmonic potential of the form Va (x) = k0 − k2 x2 + k3 x3 + k4 x4 ; (5) I. Grigorenko, M.E. Garcia / Physica A 284 (2000) 131–139 137 Fig. 4. Calculated | (x)|2 (solid line) for an electron on a potential produced by a chain of positive ions (dotted line). The inset gure shows the convergence behavior. with k0 = −137:7074997, k2 = 7, k3 = 0:5, and k4 = 1. We use these values of the parameters in order to compare with the existing calculations performed using the spectral method [20]. Our calculated ground-state energy is E0 = −144:87, whereas the value obtained by using the spectral method is E0 = −144:96., i.e., the discrepancy is less than 0:06% after 100 iterations. Our next example deals with the ground state of an electron subject to a 1D potential produced by a chain of ve positive charged ions, which is given by VQ (x) = 5 X i=1 Q ; (x − xi )2 + a2 (6) where Q is the charge of each ion and xi its position. a is a cuto distance. This smooth 1D ionic potential has been used, for instance, in the context of the Coulomb explosion of small clusters induced by intense femtosecond laser pulses [4]. In the GA-calculations for this potential, and in order to speed up the convergence process, we use for the initial populations trial functions of the form j (x) = 5 X Aj exp(−(x − xj )2 =j2 )(x − a)(b − x) ; (7) j=1 having ve peaks, where the amplitudes Aj , widths 0 ¡ j ¡ b − a and peak positions xj ∈ (a; b) are random numbers. In our calculations we used Q=5, xi =13; 19; 25; 31; 37, a = 0:5 and x ∈ [0; 50]. Note that the calculated probability distribution, shown in Fig. 4, has the same symmetry properties as the external potential VQ (x). Now we turn our attention to a many particle problem. For simplicity, we consider in this paper noninteracting fermions, but extension to interacting fermions and 138 I. Grigorenko, M.E. Garcia / Physica A 284 (2000) 131–139 Fig. 5. Calculated densities |1 (x)|2 (solid line) and |2 (x)|2 (dotted line) of the two orbitals which build the rst triplet-state wave function for two noninteracting electrons on a 1D harmonic potential (dashed line). The convergence behavior is shown in the inset. bosons can be included in our calculations in a natural way [21]. The triplet-state wave function of two noninteracting electrons √ having the lowest energy can be written as (x; x0 ) = [1 (x)2 (x0 ) − 1 (x)1 (x0 )]= 2, where 1 (x) and 2 (x) are the ground-state and rst excited state of the single-particle Hamiltonian. With the help of the GA we have determined 1 (x) and 2 (x), and consequently (x; x0 ) for the harmonic potential described above. In Fig. 5 the quantities 1 (x) and 2 (x) for this case are shown. For this calculations the individuals of the successive populations were the pairs {1 (x); 2 (x)}. Note that this procedure yields both the two-particle triplet state with the lowest energy and rst two single-particle states of the single particle Hamiltonian. For the ground-state energy the GA yields E0 = 894:90, while the analytical result is E0 = 894:43. From Figs. 1–5 it is clear, that the convergence of the GA is almost independent of the type of potential. 4. Summary and outlook In this work, we described a new powerful method for real-space computation of the ground state wave function. This approach relies on the variational principle and is based on evolutionary algorithms. The ground-state wave function of a given Hamiltonian is found by using the procedure of genetic algorithm starting from a population of trial wave functions. As was shown in this paper this method could be easy to generalize for many-body problems and more dimensions (including, for instance, the I. Grigorenko, M.E. Garcia / Physica A 284 (2000) 131–139 139 spin degrees of freedom). Results for two-dimensional systems will be described in subsequent publications [21]. It is important to point out, that it is possible to calculate not only ground state of the one particle system, but also exited states using a reformulation of this problem for N noninteracting fermions corresponding to the rst N levels [21]. Signicant improvement in the determination of the ground-state energy can be achieved by using high-order formulas for the calculation of the tness function. In most cases solution converges to an extremal value after few tens iterations, and the convergence behavior weakly depends on the complexity of the external potential. 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