PENDEKATAN FUNGSI BASIS SATU DIMENSI UNTUK MENYELESAIKAN PERSAMAAN SCHRÖDINGER UNTUK SISTEM ATOM DAN MOLEKUL MULTI-ELEKTRON
Abstract
Abstrak
Belakangan ini, kami berhasil menyelesaikan persamaan Schrödinger untuk atom multi-elektron dengan menggunakan pendekatan kombinasi linier fungsi basis satu dimensi. Implementasi fungsi basis satu dimensi memungkinkan pemisahan masalah atom menjadi komponen Cartesiannya, tidak seperti metode struktur elektronik standar yang melakukan pemisahan atas partikel. Matriks Hamiltonian dari seluruh elektron yang bersifat sparse dihasilkan dengan metode finite-difference dan pada orde standar. Kemudian sistem satu dimensi terkait dijadikan sebagai fungsi gelombang percobaan. Hasil yang didapat disempurnakan lebih lanjut dengan residual vector correction. Sebagai hasil, energi total yang diperoleh bersifat konvergen dan stabil. Total energi yang diperoleh juga akurat hingga angka desimal kelima. Selain itu, proses yang berlangsung merupakan proses variational yang berasal dari batas atas hasil dari fungsi gelombang percobaan. Dengan metode ini, banyak permasalahan, terutama yang berkaitan dengan proses evaluasi elemen matriks Hamiltonian dapat diatasi. Sebagai contoh, potensial dengan multi-center yang terdapat pada kasus molekul dapat dievaluasi dengan integrasi numerik multi-dimensi yang berlangsung dengan mudah, tanpa memerlukan pembagian sistem molekul menjadi beberapa molekul dengan pusat tunggal serta tanpa perlu dilakukan transformasi Fourier. Dibandingkan dengan pendekatan standar single-electron, solusi yang diperoleh dapat memperhitungan secara akurat efek many-body atau electron correlation yang dapat ditemukan dalam energi tolakan elektron-elektron. Terlebih, fungsi gelombang Schrödinger yang dihasilkan berisi informasi yang komprehensif yang dapat digunakan untuk melakukan plot radial correlation dan fungsi distribusi.
Kata-kata kunci: Persamaan Schrödinger, helium, molekul hidrogen, residual vector correction.
Abstract
Recently we solved the Schrödinger equation for multi-electron atoms with the use of a linear combination of one-dimensional basis functions. The implementation of the one-dimensional basis functions allows the separation of the atomic problems into their Cartesian components, unlike the standard electronic structure methods of particle-separability. The all-electron sparse Hamiltonian matrix is generated with the standard order finite-difference method, and the corresponding one-dimensional systems become the trial wave function, continued with the refinement of the results using the residual vector correction. The converged and stable energy up to five decimal places is obtained variationally from a strictly upper bound one. Many problems associated with the evaluation matrix elements of the Hamiltonian particularly the multi-center potentials present in the molecular cases are circumvented due to the easy multi-dimensional numerical integration without any partitions of molecular systems into single-center terms and any Fourier transform. Distinctive from the standard single-electron approach, the obtained solution treats more accurately many-body effect of electron correlation found in the electron-electron repulsion energy. Furthermore, the obtained Schrödinger wave function contains vast information sufficient for the radial correlation and distribution function.
Keywords: Schrödinger equation, helium, hydrogen molecule, residual vector correction, electron correlations.
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