Update index.md

Author: mtian8

docs/index.md

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 ## Introduction
-SciCode is a newly developed benchmark designed to evaluate the capabilities of language models (LMs) in generating code for solving realistic scientific research problems. It has a diverse coverage of **6** domains: Physics, Math, Material Science, Biology, and Chemistry. They span 16 diverse natural science sub-fields. Unlike previous benchmarks that consist of question-answer pairs, SciCode problems naturally factorize into multiple subproblems, each involving knowledge recall, reasoning, and code synthesis. In total, SciCode contains **338** subproblems decomposed from **80** challenging main problems, and it offers optional descriptions specifying useful scientific background information and scientist-annotated gold-standard solutions and test cases for evaluation. Claude3.5-Sonnet, the best-performing model among those tested, can solve only **4.6%** of the problems in the most realistic setting. 
+SciCode is a challenging benchmark designed to evaluate the capabilities of language models (LMs) in generating code for solving realistic scientific research problems. It has a diverse coverage of **6** domains: Physics, Math, Material Science, Biology, and Chemistry. Unlike previous benchmarks that consist of question-answer pairs, SciCode problems naturally factorize into multiple subproblems, each involving knowledge recall, reasoning, and code synthesis. In total, SciCode contains **338** subproblems decomposed from **80** challenging main problems, and it offers optional descriptions specifying useful scientific background information and scientist-annotated gold-standard solutions and test cases for evaluation. Claude3.5-Sonnet, the best-performing model among those tested, can solve only **4.6%** of the problems in the most realistic setting. 
 
 
 ## Overview
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+SciCode sources challenging and realistic research-level coding problems across 6 natural science disciplines, covering a total of 16 subfields. This diverse selection ensures a comprehensive representation of the natural sciences, where extensive code development is essential. SciCode is mainly drawn from the scripts that scientists use in their everyday workflow. Many of these have been used in one or more publications, demonstrating their robustness and correctness. Among various coding necessities, Scicode mainly focuses on 1. Numerical methods 2.Simulation of systems 3. Scientific calculation. These are the tasks we believe require intense scientific knowledge and reasoning to optimally test LM’s science capability. In designing test cases for evaluation, we incorporate domain-specific test cases in addition to numerical cases. These tests are extracted from real scientific workflows: scientists must design domain-specific test cases to verify code accuracy by reproducing results published in papers or matching analytical solutions derived from theoretical models. Each problem goes through 3 rounds of validation (i.e. by in-domain scientists, out-of-domain scientists, GPT4) for quality control.
 ![Image Title](figures/SciCode_example_problem.png)
 ## Benchmark Statistics
 
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-## Example Problem
+## Example Problem: Calculate Chern numbers for the Haldane Model
 ### Main Problem and Dependencies
 **1. Generate an array of Chern numbers for the Haldane model on a hexagonal lattice by sweeping the following parameters: the on-site energy to next-nearest-neighbor coupling constant ratio ($m/t_2$ from -6 to 6 with $N$ samples) and the phase ($\phi$ from -$\pi$ to $\pi$ with $N$ samples) values. Given the lattice spacing $a$, the nearest-neighbor coupling constant $t_1$, the next-nearest-neighbor coupling constant $t_2$, the grid size $\delta$ for discretizing the Brillouin zone in the $k_x$ and $k_y$ directions (assuming the grid sizes are the same in both directions), and the number of sweeping grid points $N$ for $m/t_2$ and $\phi$.**